The Riemann–Hurwitz formula is the calculation of the Euler characteristic of the branched covering in terms of the ramification indices and degree of the cover. If $\tilde{\pi} := \pi \circ n : \tilde{C} \to \mathbb{CP}^1$ is a k-fold ramified cover with ramification points $x_1, \dots, x_\ell$ with corresponding ramification indices $e_{\tilde{\pi}}(x_j)$, then

$$ \chi(\tilde{C}) = k(2 - \ell) + \sum_{j=1}^{\ell} (k + 1 - e_{\tilde{\pi}}(x_j)) = 2k - \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1). $$

In our case, $\tilde{C}$ is a 2-sphere so $\chi(\tilde{C}) = 2$. Suppose $d$ is the degree of $C$. If we choose $p \notin C$, then a generic line through $p$ intersects $C$ $d$ times, so the degree of the cover is $d$. Therefore, this equation specializes to

$$ 2d - 2 = \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1). $$

If instead, we choose $p \in C$, where $p$ has multiplicity $m_p$ (where $m_p$ is the first entry of the multiplicity sequence if $p$ is a singular point and is 1 if $p$ is a smooth point of $C$), then a generic line through $p$ intersects $C$ at $d - m_p$ other points. Therefore, $\tilde{\pi}$ is a $(d - m_p)$-fold cover. This gives the following equation and inequality. The inequality is particularly useful as an obstruction to symplectically realizing certain cuspidal curves in $\mathbb{CP}^2$:

$$ 2(d - m_p) - 2 = \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1) \geq \sum_{q \neq p} (m_q - 1) + (m_{p,2} - 1), $$

which we re-write as follows,

$$ 2d - 2m_p \geq 2 + \sum_{q \neq p} (m_q - 1) + (m_{p,2} - 1). \qquad (3.1) $$

*Example 3.20.* We can apply Riemann–Hurwitz to exclude the following configurations of cusps for a quintic: $\{[3, 2], [2], [2]\}$, $\{[3], [2], [2], [2]\}$, $\{[2, 2], [2], [2], [2], [2]\}$, and $\{[2], [2], [2], [2], [2], [2]\}$. In the first case, we project from the $[3, 2]$-cusp, and we obtain the following contradiction:

$$ 2 \cdot 5 - 2 \cdot 3 \geq 2 + 1 + 1 + 1. $$

In the second, we project from the $[3]$-cusp:

$$ 2 \cdot 5 - 2 \cdot 3 \geq 2 + 1 + 1 + 1. $$

In the third case, we project from the $[2, 2]$-cusp:

$$ 2 \cdot 5 - 2 \cdot 2 \geq 2 + 1 + 1 + 1 + 1 + 1, $$

and in the last from any of the cusps, to obtain

$$ 2 \cdot 5 - 2 \cdot 2 \geq 2 + 1 + 1 + 1 + 1 + 1. $$

Thus, in each of the four cases, we get a contradiction.

## 4. The necessity of rationality

An interesting question to ask is the following: can we distinguish exotic symplectic 4-manifolds (for example, a potentially exotic $\mathbb{CP}^2$) in terms of the singular symplectic submanifolds they contain? For example, we show that there are certain rational cuspidal curves which admit no symplectic embedding in $\mathbb{CP}^2$, but a priori such curves could admit symplectic embeddings in
The Riemann–Hurwitz formula is the calculation of the Euler characteristic of the branched covering in terms of the ramification indices and degree of the cover. If $\tilde{\pi} := \pi \circ n : \tilde{C} \to \mathbb{CP}^1$ is a k-fold ramified cover with ramification points $x_1, \dots, x_\ell$ with corresponding ramification indices $e_{\tilde{\pi}}(x_j)$, then

$$ \chi(\tilde{C}) = k(2 - \ell) + \sum_{j=1}^{\ell} (k + 1 - e_{\tilde{\pi}}(x_j)) = 2k - \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1). $$

In our case, $\tilde{C}$ is a 2-sphere so $\chi(\tilde{C}) = 2$. Suppose $d$ is the degree of $C$. If we choose $p \notin C$, then a generic line through $p$ intersects $C$ $d$ times, so the degree of the cover is $d$. Therefore, this equation specializes to

$$ 2d - 2 = \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1). $$

If instead, we choose $p \in C$, where $p$ has multiplicity $m_p$ (where $m_p$ is the first entry of the multiplicity sequence if $p$ is a singular point and is 1 if $p$ is a smooth point of $C$), then a generic line through $p$ intersects $C$ at $d - m_p$ other points. Therefore, $\tilde{\pi}$ is a $(d - m_p)$-fold cover. This gives the following equation and inequality. The inequality is particularly useful as an obstruction to symplectically realizing certain cuspidal curves in $\mathbb{CP}^2$:

$$ 2(d - m_p) - 2 = \sum_{j=1}^{\ell} (e_{\tilde{\pi}}(x_j) - 1) \geq \sum_{q \neq p} (m_q - 1) + (m_{p,2} - 1), $$

which we re-write as follows,

$$ 2d - 2m_p \geq 2 + \sum_{q \neq p} (m_q - 1) + (m_{p,2} - 1). \quad (3.1) $$

*Example 3.20.* We can apply Riemann–Hurwitz to exclude the following configurations of cusps for a quintic: $\{[3, 2], [2], [2]\}$, $\{[3], [2], [2], [2]\}$, $\{[2, 2], [2], [2], [2], [2]\}$, and $\{[2], [2], [2], [2], [2], [2]\}$. In the first case, we project from the $[3, 2]$-cusp, and we obtain the following contradiction:

$$ 2 \cdot 5 - 2 \cdot 3 \geq 2 + 1 + 1 + 1. $$

In the second, we project from the $[3]$-cusp:

$$ 2 \cdot 5 - 2 \cdot 3 \geq 2 + 1 + 1 + 1. $$

In the third case, we project from the $[2, 2]$-cusp:

$$ 2 \cdot 5 - 2 \cdot 2 \geq 2 + 1 + 1 + 1 + 1 + 1, $$

and in the last from any of the cusps, to obtain

$$ 2 \cdot 5 - 2 \cdot 2 \geq 2 + 1 + 1 + 1 + 1 + 1. $$

Thus, in each of the four cases, we get a contradiction.

## 4. The necessity of rationality

An interesting question to ask is the following: can we distinguish exotic symplectic 4-manifolds (for example, a potentially exotic $\mathbb{CP}^2$) in terms of the singular symplectic submanifolds they contain? For example, we show that there are certain rational cuspidal curves which admit no symplectic embedding in $\mathbb{CP}^2$, but a priori such curves could admit symplectic embeddings in
an exotic symplectic $\mathbb{CP}^2$. Though this is an alluring hope, we explain here that rational cuspidal curves we consider here cannot exist in any exotic or even homology $\mathbb{CP}^2$. As mentioned in the introduction, a similar question was raised, and partially addressed, by Chen [Che18].

We recall that Taubes proved that any symplectic structure on the standard, smooth $\mathbb{CP}^2$ is in fact symplectomorphic, up to rescaling, to the Fubini-Study form [Tau96, Theorem 0.3].

We consider triples $(X, \omega, C)$, where $X$ is a rational homology $\mathbb{CP}^2$, $\omega$ is a symplectic form on $X$, and $C \subset X$ is a symplectic rational cuspidal curve, and show that $X$ is necessarily $\mathbb{CP}^2$.

This question is answered in the algebraic setting by dividing into two cases, depending on the sign of the canonical divisor. If $X$ is an algebraic surface that is a rational homology $\mathbb{CP}^2$, either $K_X$ is not nef, which proves that $X$ is rational (and hence $\mathbb{CP}^2$, by our homological assumption) or $K_X$ is nef, in which case Yau proved that $X$ is a ball quotient [Aub76, Yau77]. These are known as Mumford surfaces, or fake projective planes; the first example was given by Mumford [Mum79], and were classified by Cartwright and Steger [CS10], building on work of Prasad and Yeung [PY07]. As they are all ball quotients, they have no symplectic rational curves, for maps from $S^2$ lift to the universal cover, and there are no compact complex (or symplectic) curves in $B^4$.

In the proof of Theorem 1.4 we also distinguish between the two cases, according to the sign of the canonical divisor $K_X$ (or, equivalently, of the first Chern class); we use techniques inspired by gauge theory in both cases, albeit in two different ways. The proof in the case where $K_X < 0$ is essentially known to experts, and was already proved by Chen [Che18, Corollary 2.3]; we give a slightly different proof here. The proof in the case $K_X > 0$ uses tools from Heegaard Floer homology, and is the part of the proof that is genuinely novel.

*Proof of Theorem 1.4.* Recall that, for links of curve singularities and their connected sums one has that the invariant $\nu^+$ coming from Heegaard Floer homology [HW16], the slice genus, and the 3-genus all agree [HW16, Proposition 3]. As $p_g(C) = 0$, the arithmetic genus $p_a(C)$ of $C$ is equal to the sum of the $\delta$-invariants of the singular points of $C$. In particular, if we let $K$ denote the connected sum of all links of singularities of $C$, then $\nu^+(K) = p_a(C)$.

As $X$ admits a compatible almost-complex structure $J$, $c_1^2(J) - 2\chi(X) - 3\sigma(X) = 0$. As $X$ is assumed to be a homotopy $\mathbb{CP}^2$, this implies that $c_1^2(J) = 9$ and, thus, $c_1(J) = \pm 3 \mathrm{PD}(h)$ (where $h = [\mathbb{CP}^1] \in H_2(\mathbb{CP}^2; \mathbb{Z}))$.

If $C$ has degree $d$ and $c_1(J) = 3h$, as it is for $(\mathbb{CP}^2, \omega_{\mathrm{FS}})$, the adjunction formula (2.2) yields $p_a(C) = \frac{1}{2}(d-1)(d-2)$, and Proposition 4.1 guarantees that $X$ is the standard $\mathbb{CP}^2$.

We turn now to the case when $c_1(J) = -3h$; as in the algebraic case, we argue that there are no symplectic 4-manifolds $X$ with $c_1(J) = -3h$ admitting rational cuspidal curves. In fact, now the adjunction formula yields

$$p_a(C) = \frac{(d+1)(d+2)}{2}.$$

Consider the boundary of a regular neighborhood $N$ of $C$, which is homeomorphic to $S_{d_2}^3(K)$. Then the complement of the interior of $N$, taken with the opposite orientation, is a rational homology ball whose boundary is $S_{d_2}^3(K)$. Since $p_a(C) = \nu^+(K)$ we have that

$$\frac{d(d-1)}{2} - \nu^+(K) = \frac{d(d-1) - (d+1)(d+2)}{2} = -2d - 1 < 0,$$

contradicting the bound obtained in [AG17, Theorem 5.1]. □

We conclude the proof with the following proposition, which is mostly a corollary of [MS96, Corollary 1.5]. This, in turn, follows from work of Liu [Liu96], building on Taubes' work on
an exotic symplectic $\mathbb{CP}^2$. Though this is an alluring hope, we explain here that rational cuspidal curves we consider here cannot exist in any exotic or even homology $\mathbb{CP}^2$. As mentioned in the introduction, a similar question was raised, and partially addressed, by Chen [Che18].

We recall that Taubes proved that any symplectic structure on the standard, smooth $\mathbb{CP}^2$ is in fact symplectomorphic, up to rescaling, to the Fubini-Study form [Tau96, Theorem 0.3].

We consider triples $(X, \omega, C)$, where $X$ is a rational homology $\mathbb{CP}^2$, $\omega$ is a symplectic form on $X$, and $C \subset X$ is a symplectic rational cuspidal curve, and show that $X$ is necessarily $\mathbb{CP}^2$.

This question is answered in the algebraic setting by dividing into two cases, depending on the sign of the canonical divisor. If $X$ is an algebraic surface that is a rational homology $\mathbb{CP}^2$, either $K_X$ is not nef, which proves that $X$ is rational (and hence $\mathbb{CP}^2$, by our homological assumption) or $K_X$ is nef, in which case Yau proved that $X$ is a ball quotient [Aub76, Yau77]. These are known as Mumford surfaces, or fake projective planes; the first example was given by Mumford [Mum79], and were classified by Cartwright and Steger [CS10], building on work of Prasad and Yeung [PY07]. As they are all ball quotients, they have no symplectic rational curves, for maps from $S^2$ lift to the universal cover, and there are no compact complex (or symplectic) curves in $B^4$.

In the proof of Theorem 1.4 we also distinguish between the two cases, according to the sign of the canonical divisor $K_X$ (or, equivalently, of the first Chern class); we use techniques inspired by gauge theory in both cases, albeit in two different ways. The proof in the case where $K_X < 0$ is essentially known to experts, and was already proved by Chen [Che18, Corollary 2.3]; we give a slightly different proof here. The proof in the case $K_X > 0$ uses tools from Heegaard Floer homology, and is the part of the proof that is genuinely novel.

*Proof of Theorem 1.4.* Recall that, for links of curve singularities and their connected sums one has that the invariant $\nu^+$ coming from Heegaard Floer homology [HW16], the slice genus, and the 3-genus all agree [HW16, Proposition 3]. As $p_g(C) = 0$, the arithmetic genus $p_a(C)$ of $C$ is equal to the sum of the $\delta$-invariants of the singular points of $C$. In particular, if we let $K$ denote the connected sum of all links of singularities of $C$, then $\nu^+(K) = p_a(C)$.

As $X$ admits a compatible almost-complex structure $J$, $c_1^2(J) - 2\chi(X) - 3\sigma(X) = 0$. As $X$ is assumed to be a homotopy $\mathbb{CP}^2$, this implies that $c_1^2(J) = 9$ and, thus, $c_1(J) = \pm 3 \mathrm{PD}(h)$ (where $h = [\mathbb{CP}^1] \in H_2(\mathbb{CP}^2; \mathbb{Z}))$.

If $C$ has degree $d$ and $c_1(J) = 3h$, as it is for $(\mathbb{CP}^2, \omega_{\mathrm{FS}})$, the adjunction formula (2.2) yields $p_a(C) = \frac{1}{2}(d-1)(d-2)$, and Proposition 4.1 guarantees that $X$ is the standard $\mathbb{CP}^2$.

We turn now to the case when $c_1(J) = -3h$; as in the algebraic case, we argue that there are no symplectic 4-manifolds $X$ with $c_1(J) = -3h$ admitting rational cuspidal curves. In fact, now the adjunction formula yields

$$p_a(C) = \frac{(d+1)(d+2)}{2}.$$

Consider the boundary of a regular neighborhood $N$ of $C$, which is homeomorphic to $S_{d2}^3(K)$. Then the complement of the interior of $N$, taken with the opposite orientation, is a rational homology ball whose boundary is $S_{d2}^3(K)$. Since $p_a(C) = \nu^+(K)$ we have that

$$\frac{d(d-1)}{2} - \nu^+(K) = \frac{d(d-1) - (d+1)(d+2)}{2} = -2d - 1 < 0,$$

contradicting the bound obtained in [AG17, Theorem 5.1]. □

We conclude the proof with the following proposition, which is mostly a corollary of [MS96, Corollary 1.5]. This, in turn, follows from work of Liu [Liu96], building on Taubes' work on
Seiberg-Witten theory. (See also [Wen18, Theorem 7.36] for a more modern and self-contained treatment.) The theorem asserts that, if $C$ is a smooth symplectic curve in a closed symplectic 4-manifold $(X, \omega)$ such that $\langle c_1(\omega), C \rangle \ge 1$, and $C$ is not a $(-1)$-sphere, then $(X, \omega)$ is rational or ruled. Using the adjunction formula, we can rephrase the hypothesis on the first Chern class into an assumption on the self-intersection. Namely $\langle c_1(\omega), C \rangle = 2 - 2g(C) + [C]^2$, so the hypothesis is equivalent to $[C]^2 \ge 2g(\Sigma) - 1$ and $C$ is not an exceptional sphere. (Note that the case $g(\Sigma) = 0$ is Theorem 3.6.)

**PROPOSITION 4.1.** Suppose $C$ is a rational cuspidal curve with $C \cdot C \ge 2p_a(C) - 1 > 0$. If $C$ symplectically embeds into a closed symplectic manifold $(X, \omega)$, then $(X, \omega)$ is a rational surface.

*Proof.* By deforming the curve $C$ in a regular neighborhood using the Milnor fibration model, we can find a *smooth*, symplectic surface $C' \subset (X, \omega)$ in the same homology class as $C$, with genus $g(C') = p_a(C)$. By [MS96, Corollary 1.5], $(X, \omega)$ must be either a rational surface or an irrational ruled surface $(X', \omega')$. However, Lemma 4.2 shows that $X'$ cannot be irrational ruled. $\square$

**LEMMA 4.2.** Let $C$ be a (possibly singular) curve of positive self-intersection in a (possibly non-minimal) symplectic 4-manifold $(X', \omega')$, ruled over a Riemann surface $\Sigma$. Then $p_g(C) \ge g(\Sigma)$.

*Proof.* Let $\tilde{J} \in \mathcal{J}^{\omega'}(C)$. By [McD90, Theorem 3.4] we can find a maximal collection of disjoint $\tilde{J}$-holomorphic exceptional spheres, to blow down to a minimal surface $b : (X', \omega') \to (X, \omega)$ that is still ruled over $\Sigma$. Note that $b(C)$ is still a singular symplectic surface and it is $J$-holomorphic where $J$ is the almost complex structure on $X$ induced from $\tilde{J}$.

Now, $(X, \omega)$ is symplectomorphic to a ruled surface. Let $B$ be the homology class of a smooth fiber. Consider the moduli space $\mathcal{M}(J', B)$. As argued in [McD90, Proposition 4.1], if $J$ is regular for the associated Fredholm operator, there is a unique $J$-holomorphic $B$ curve from $\mathcal{M}(J, B)$ through each point $p \in X$. Moreover, because $B^2 = 0 > -2$, $J$ is automatically a regular value of the associated Fredholm operator $P_B$ (see [Gro85], [McD90, Lemma 2.8], and [HLS97]). Therefore, we have a projection map $\pi : X \to \Sigma$ whose fibers intersect $b(C)$ positively.

By compactness and positivity of intersection, $(\pi \circ b)|_C$ is onto $\Sigma$. Let $n : C' \to C$ be the normalization map, and note that, by definition, $g(C') = p_g(C)$. Pre-composing $\pi \circ b$ with $n$ yields a surjective map $\pi' = (\pi \circ b) \circ n : C' \to \Sigma$, hence $g(C') \ge g(\Sigma)$, as desired. $\square$

# 5. Symplectic isotopy problems for reducible configurations

In order to obstruct and classify symplectic isotopy classes of rational cuspidal curves and symplectic fillings of their associated contact manifolds, we perform birational transformations to relate the original problem to a classification of a reducible configuration consisting of multiple lower-degree smooth components intersecting in a particular way. Depending on the intersection pattern, such a configuration may be verified to have a unique symplectic isotopy class, or shown not to exist in $\mathbb{CP}^2$. Examples of such configurations shown not to exist symplectically in $\mathbb{CP}^2$ appeared in [RS19]. The example from that article that we use most is the Fano plane: a configuration of seven lines intersecting in seven transverse triple points.

Typically, we use script letters to denote abstract configuration types, and non-script letters to denote actual realizations.

## 5.1 Existence and uniqueness

Our first method of extending symplectic isotopy results to a larger collection of reducible configurations is through the following lemma that allows us to add a line with limited constraints
Seiberg-Witten theory. (See also [Wen18, Theorem 7.36] for a more modern and self-contained treatment.) The theorem asserts that, if $C$ is a smooth symplectic curve in a closed symplectic 4-manifold $(X, \omega)$ such that $\langle c_1(\omega), C \rangle \ge 1$, and $C$ is not a $(-1)$-sphere, then $(X, \omega)$ is rational or ruled. Using the adjunction formula, we can rephrase the hypothesis on the first Chern class into an assumption on the self-intersection. Namely $\langle c_1(\omega), C \rangle = 2 - 2g(C) + [C]^2$, so the hypothesis is equivalent to $[C]^2 \ge 2g(\Sigma) - 1$ and $C$ is not an exceptional sphere. (Note that the case $g(\Sigma) = 0$ is Theorem 3.6.)

**PROPOSITION 4.1.** Suppose $C$ is a rational cuspidal curve with $C \cdot C \ge 2p_a(C) - 1 > 0$. If $C$ symplectically embeds into a closed symplectic manifold $(X, \omega)$, then $(X, \omega)$ is a rational surface.

*Proof.* By deforming the curve $C$ in a regular neighborhood using the Milnor fibration model, we can find a *smooth*, symplectic surface $C' \subset (X, \omega)$ in the same homology class as $C$, with genus $g(C') = p_a(C)$. By [MS96, Corollary 1.5], $(X, \omega)$ must be either a rational surface or an irrational ruled surface $(X', \omega')$. However, Lemma 4.2 shows that $X'$ cannot be irrational ruled. $\square$

**LEMMA 4.2.** Let $C$ be a (possibly singular) curve of positive self-intersection in a (possibly non-minimal) symplectic 4-manifold $(X', \omega')$, ruled over a Riemann surface $\Sigma$. Then $p_g(C) \ge g(\Sigma)$.

*Proof.* Let $\tilde{J} \in \mathcal{J}^{\omega'}(C)$. By [McD90, Theorem 3.4] we can find a maximal collection of disjoint $\tilde{J}$-holomorphic exceptional spheres, to blow down to a minimal surface $b: (X', \omega') \to (X, \omega)$ that is still ruled over $\Sigma$. Note that $b(C)$ is still a singular symplectic surface and it is $J$-holomorphic where $J$ is the almost complex structure on $X$ induced from $\tilde{J}$.

Now, $(X, \omega)$ is symplectomorphic to a ruled surface. Let $B$ be the homology class of a smooth fiber. Consider the moduli space $\mathcal{M}(J', B)$. As argued in [McD90, Proposition 4.1], if $J$ is regular for the associated Fredholm operator, there is a unique $J$-holomorphic $B$ curve from $\mathcal{M}(J, B)$ through each point $p \in X$. Moreover, because $B^2 = 0 > -2$, $J$ is automatically a regular value of the associated Fredholm operator $P_B$ (see [Gro85], [McD90, Lemma 2.8], and [HLS97]). Therefore, we have a projection map $\pi: X \to \Sigma$ whose fibers intersect $b(C)$ positively.

By compactness and positivity of intersection, $(\pi \circ b)|_C$ is onto $\Sigma$. Let $n: C' \to C$ be the normalization map, and note that, by definition, $g(C') = p_g(C)$. Pre-composing $\pi \circ b$ with $n$ yields a surjective map $\pi' = (\pi \circ b) \circ n: C' \to \Sigma$, hence $g(C') \ge g(\Sigma)$, as desired. $\square$

# 5. Symplectic isotopy problems for reducible configurations

In order to obstruct and classify symplectic isotopy classes of rational cuspidal curves and symplectic fillings of their associated contact manifolds, we perform birational transformations to relate the original problem to a classification of a reducible configuration consisting of multiple lower-degree smooth components intersecting in a particular way. Depending on the intersection pattern, such a configuration may be verified to have a unique symplectic isotopy class, or shown not to exist in $\mathbb{CP}^2$. Examples of such configurations shown not to exist symplectically in $\mathbb{CP}^2$ appeared in [RS19]. The example from that article that we use most is the Fano plane: a configuration of seven lines intersecting in seven transverse triple points.

Typically, we use script letters to denote abstract configuration types, and non-script letters to denote actual realizations.

## 5.1 Existence and uniqueness

Our first method of extending symplectic isotopy results to a larger collection of reducible configurations is through the following lemma that allows us to add a line with limited constraints
to an existing configuration. A single smooth component is known to have a unique symplectic isotopy class in $\mathbb{CP}^2$ if its degree is at most 17 (see [ST05]).

The following proposition gives a way of obtaining many configurations in $\mathbb{CP}^2$ with unique isotopy classes by adding degree-one components sufficiently generically.

All configurations in the statements have labeled components and labeled singular points, and isotopies preserve the labelings. That is, if $\mathcal{C}$ is a configuration of $n$ curves, then its components are labeled $B_1, \dots, B_n$, and its singular points are labeled $P^1, \dots, P^m$. When we say that two realizations of $\mathcal{C}$ are isotopic, we mean that the isotopy preserves the labeling. Viewed differently, an isotopy from a labeled realization of $\mathcal{C}$ to an unlabeled one induces a labeling of $\mathcal{C}$.

**PROPOSITION 5.1.** Suppose $C_1$ is a configuration of singular symplectic curves in $(\mathbb{CP}^2, \omega_{FS})$ obtained from $C_0$ by adding a single symplectic line $L$. Suppose that in the configuration $C_1$ either:

(i) $L$ intersects the curves of $C_0$ transversally and the intersection points of $L$ with $C_0$ contain at most two singular points, $P^i$ and $P^j$, of $C_0$; or

(ii) $L$ has a simple tangency to a subset of the components of $C_0$, $\{B_{i_1}, \dots, B_{i_k}\}$ at a single point $P^i$ in $C_0$ ($P^i$ may be either a smooth or singular point of $B_\ell$ and it can be a singular point of $C_0$ in which case it uses the existing label, or a smooth point of $C_0$ in which case it takes a new label index) and all other intersections of $L$ with $C_0$ are transverse double points; further assume in this case that in $C_1$, there are no other intersections of $L$ with the components $B_{i_1}, \dots, B_{i_k}$ outside of the tangent point (but $L$ may intersect other components in transverse double points).

Then there is a bijection between the isotopy classes of realizations of $C_0$ and those of $C_1$. In particular, $C_0$ has a unique equisingular symplectic isotopy class if and only if $C_1$ does.

The hypothesis in item (ii) fixes the multiplicity of the intersection of $L$ with the components of $C_0$ at the point $P$. The requirement that the tangency is simple means that the multiplicity of the intersection is as small as possible for a symplectic curve with tangential components. More specifically, when $B_\ell$ is smooth at $P$, a line has a simple tangency with $B_\ell$ if and only if the multiplicity of intersection is two. If $B_\ell$ has a singularity at $P$, a line has a simple tangency to $B_\ell$ if and only if the multiplicity of intersection is equal to the third element of the semigroup of $(B_\ell, P)$. The second requirement in the hypothesis in item (ii) says that the multiplicity of intersection of $L$ with $B_\ell$ at $P$ is as large as it possibly can be for global degree reasons. Thus, these two conditions ensure that the multiplicity of intersection between $L$ and $B_\ell$ is automatically as it should be whenever $L$ is tangent to $B_\ell$ and their union is a singular symplectic configuration.

Note that the second constraint in item (ii) is automatically satisfied when $B_\ell$ is a smooth conic (degree-two curve). It also holds when the tangency occurs at a singular point of $B_\ell$ with multiplicity sequence $[d-1]$ when $B$ has degree $d$. This will be sufficient to cover our applications in this article. We hope to generalize this isotopy statement for reducible configurations and in particular, remove the second constraint from the hypothesis in item (ii) through future work.

In most cases, in this article we need not worry about the labels: in most cases, the type of the points $P^i$ and $P^j$ (or, in case (ii), of $P^i$) determine the labeling. (For instance, there could be exactly two triple points in the configuration $C_0$, so that ‘the line passing through the two triple points’ is well defined, or a unique conic in $C_0$, so that ‘the line tangent to the conic at a generic point’ is well defined.) However, in general the labels are important when applying this
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

to an existing configuration. A single smooth component is known to have a unique symplectic isotopy class in $\mathbb{CP}^2$ if its degree is at most 17 (see [ST05]).

The following proposition gives a way of obtaining many configurations in $\mathbb{CP}^2$ with unique isotopy classes by adding degree-one components sufficiently generically.

All configurations in the statements have labeled components and labeled singular points, and isotopies preserve the labelings. That is, if $\mathcal{C}$ is a configuration of $n$ curves, then its components are labeled $B_1, \dots, B_n$, and its singular points are labeled $P^1, \dots, P^m$. When we say that two realizations of $\mathcal{C}$ are isotopic, we mean that the isotopy preserves the labeling. Viewed differently, an isotopy from a labeled realization of $\mathcal{C}$ to an unlabeled one induces a labeling of $\mathcal{C}$.

**PROPOSITION 5.1.** Suppose $C_1$ is a configuration of singular symplectic curves in $(\mathbb{CP}^2, \omega_{FS})$ obtained from $C_0$ by adding a single symplectic line $L$. Suppose that in the configuration $C_1$ either:

(i) $L$ intersects the curves of $C_0$ transversally and the intersection points of $L$ with $C_0$ contain at most two singular points, $P^i$ and $P^j$, of $C_0$; or

(ii) $L$ has a simple tangency to a subset of the components of $C_0$, $\{B_{i_1}, \dots, B_{i_k}\}$ at a single point $P^i$ in $C_0$ ($P^i$ may be either a smooth or singular point of $B_\ell$ and it can be a singular point of $C_0$ in which case it uses the existing label, or a smooth point of $C_0$ in which case it takes a new label index) and all other intersections of $L$ with $C_0$ are transverse double points; further assume in this case that in $C_1$, there are no other intersections of $L$ with the components $B_{i_1}, \dots, B_{i_k}$ outside of the tangent point (but $L$ may intersect other components in transverse double points).

Then there is a bijection between the isotopy classes of realizations of $C_0$ and those of $C_1$. In particular, $C_0$ has a unique equisingular symplectic isotopy class if and only if $C_1$ does.

The hypothesis in item (ii) fixes the multiplicity of the intersection of $L$ with the components of $C_0$ at the point $P$. The requirement that the tangency is simple means that the multiplicity of the intersection is as small as possible for a symplectic curve with tangential components. More specifically, when $B_\ell$ is smooth at $P$, a line has a simple tangency with $B_\ell$ if and only if the multiplicity of intersection is two. If $B_\ell$ has a singularity at $P$, a line has a simple tangency to $B_\ell$ if and only if the multiplicity of intersection is equal to the third element of the semigroup of $(B_\ell, P)$. The second requirement in the hypothesis in item (ii) says that the multiplicity of intersection of $L$ with $B_\ell$ at $P$ is as large as it possibly can be for global degree reasons. Thus, these two conditions ensure that the multiplicity of intersection between $L$ and $B_\ell$ is automatically as it should be whenever $L$ is tangent to $B_\ell$ and their union is a singular symplectic configuration.

Note that the second constraint in item (ii) is automatically satisfied when $B_\ell$ is a smooth conic (degree-two curve). It also holds when the tangency occurs at a singular point of $B_\ell$ with multiplicity sequence $[d-1]$ when $B$ has degree $d$. This will be sufficient to cover our applications in this article. We hope to generalize this isotopy statement for reducible configurations and in particular, remove the second constraint from the hypothesis in item (ii) through future work.

In most cases, in this article we need not worry about the labels: in most cases, the type of the points $P^i$ and $P^j$ (or, in case (ii), of $P^i$) determine the labeling. (For instance, there could be exactly two triple points in the configuration $C_0$, so that ‘the line passing through the two triple points’ is well defined, or a unique conic in $C_0$, so that ‘the line tangent to the conic at a generic point’ is well defined.) However, in general the labels are important when applying this
<figure><img src="image_13.png" /><figcaption>FIGURE 4. Two configurations $\mathcal{B}_1$ and $\mathcal{B}_2$ which are equivalent as unlabeled configurations. The circle and oval represent conics $C_1$ and $C_2$. The vertical lines represent lines $L_1$ and $L_2$. A third line $L$ is the diagonal line in $\mathcal{B}_1$ and the horizontal line in $\mathcal{B}_2$. The configurations $\mathcal{B}_1$ and $\mathcal{B}_2$ are combinatorially equivalent as unlabeled configurations.</figcaption></figure>

proposition. Specifically there may be more than one point in the configuration $C_0$ of the same type (even lying on the same collection of curves), which one may choose as candidates for $P^i$ (or $P^j$). In this case, the configuration $C_1$ obtained from one choice of candidate for $P^i$ is different than the configuration $C'_1$ obtained from a different choice of candidate for $P^i$, and there need not be an equisingular isotopy which takes a realization of $C_1$ to a realization of $C'_1$, even if $C_0$ has a unique equisingular isotopy class.

*Example 5.2.* This phenomenon is made explicit in [ASST21]. The configurations concerned, $\mathcal{B}_1$ and $\mathcal{B}_2$, are made up of two conics $C_1$ and $C_2$ which are tangent at two points, two generic tangents $L_1$ and $L_2$ to $C_1$, and a third line $L$ passing through a point in $L_1 \cap C_2$ and through a point in $L_2 \cap C_2$. Depending on the choice of the point in $L_2 \cap C_2$ we obtain two possible configurations as in Figure 4 (the line $L$ can be either the diagonal line or the horizontal line). As unlabeled abstract configurations, $\mathcal{B}_1$ and $\mathcal{B}_2$ are the same; as a labeled configuration, there are (a priori) four choices for $L$. It is shown in [ASST21] that two of the four choices admit realizations $\mathcal{B}_1$ and $\mathcal{B}_2$ whose complements have non-isomorphic fundamental groups. (In fact, it is clear from the figure that choices come in symmetric pairs.)

We now can see that an unlabeled version of Proposition 5.1 would lead to a contradiction. Here $\mathcal{B}_i \setminus L$ has a unique isotopy class: this can be seen, for instance, using the techniques we develop in this section to show that the two conics have a unique isotopy class and then applying Proposition 5.1 twice. If we tried to apply an unlabeled version of Proposition 5.1, we would treat $\mathcal{B}_1$ and $\mathcal{B}_2$ as the same configuration, and thus their realizations $\mathcal{B}_1$ and $\mathcal{B}_2$ of Figure 4 (which is taken from [ASST21, Figure 1]) would be symplectically isotopic. As we know by [ASST21] that these are non-isotopic, we see that this unlabeled version of Proposition 5.1 is false.

*Proof.* Let $\text{Iso}(C_0)$ and $\text{Iso}(C_1)$ denote the sets of symplectic isotopy classes in $(\mathbb{CP}^2, \omega_{\text{FS}})$ of the configurations $C_0$ and $C_1$, respectively. There is a natural map

$$ \Psi : \text{Iso}(C_1) \to \text{Iso}(C_0) $$

defined by $\Psi([C']) = [C]$ where $C$ is the realization of $C_0$ obtained from the realization $C'$ of $C_1$ by deleting the realization in $C'$ of the line $L$. It is clear that this map is well-defined, because if $C'_1$ and $C'_2$ are symplectically isotopic realizations of $C_1$, there exists a symplectic isotopy $C'_t$
<figure><img src="image_13.png" /><figcaption>FIGURE 4. Two configurations $\mathcal{B}_1$ and $\mathcal{B}_2$ which are equivalent as unlabeled configurations. The circle and oval represent conics $C_1$ and $C_2$. The vertical lines represent lines $L_1$ and $L_2$. A third line $L$ is the diagonal line in $\mathcal{B}_1$ and the horizontal line in $\mathcal{B}_2$. The configurations $\mathcal{B}_1$ and $\mathcal{B}_2$ are combinatorially equivalent as unlabeled configurations.</figcaption></figure>

proposition. Specifically there may be more than one point in the configuration $C_0$ of the same type (even lying on the same collection of curves), which one may choose as candidates for $P^i$ (or $P^j$). In this case, the configuration $C_1$ obtained from one choice of candidate for $P^i$ is different than the configuration $C'_1$ obtained from a different choice of candidate for $P^i$, and there need not be an equisingular isotopy which takes a realization of $C_1$ to a realization of $C'_1$, even if $C_0$ has a unique equisingular isotopy class.

Example 5.2. This phenomenon is made explicit in [ASST21]. The configurations concerned, $\mathcal{B}_1$ and $\mathcal{B}_2$, are made up of two conics $C_1$ and $C_2$ which are tangent at two points, two generic tangents $L_1$ and $L_2$ to $C_1$, and a third line $L$ passing through a point in $L_1 \cap C_2$ and through a point in $L_2 \cap C_2$. Depending on the choice of the point in $L_2 \cap C_2$ we obtain two possible configurations as in Figure 4 (the line $L$ can be either the diagonal line or the horizontal line). As unlabeled abstract configurations, $\mathcal{B}_1$ and $\mathcal{B}_2$ are the same; as a labeled configuration, there are (a priori) four choices for $L$. It is shown in [ASST21] that two of the four choices admit realizations $\mathcal{B}_1$ and $\mathcal{B}_2$ whose complements have non-isomorphic fundamental groups. (In fact, it is clear from the figure that choices come in symmetric pairs.)

We now can see that an unlabeled version of Proposition 5.1 would lead to a contradiction. Here $\mathcal{B}_i \setminus L$ has a unique isotopy class: this can be seen, for instance, using the techniques we develop in this section to show that the two conics have a unique isotopy class and then applying Proposition 5.1 twice. If we tried to apply an unlabeled version of Proposition 5.1, we would treat $\mathcal{B}_1$ and $\mathcal{B}_2$ as the same configuration, and thus their realizations $\mathcal{B}_1$ and $\mathcal{B}_2$ of Figure 4 (which is taken from [ASST21, Figure 1]) would be symplectically isotopic. As we know by [ASST21] that these are non-isotopic, we see that this unlabeled version of Proposition 5.1 is false.

*Proof.* Let $\text{Iso}(C_0)$ and $\text{Iso}(C_1)$ denote the sets of symplectic isotopy classes in $(\mathbb{CP}^2, \omega_{\text{FS}})$ of the configurations $C_0$ and $C_1$, respectively. There is a natural map

$$ \Psi: \text{Iso}(C_1) \to \text{Iso}(C_0) $$

defined by $\Psi([C']) = [C]$ where $C$ is the realization of $C_0$ obtained from the realization $C'$ of $C_1$ by deleting the realization in $C'$ of the line $L$. It is clear that this map is well-defined, because if $C'_1$ and $C'_2$ are symplectically isotopic realizations of $C_1$, there exists a symplectic isotopy $C'_t$
between them and dropping the realization of the line $L$ from each $C'_t$ yields a symplectic isotopy between the realizations $C_1$ and $C_2$ of $C_0$. We show that the map $\Psi$ is surjective and injective to obtain the stated result.

Recall that Gromov proved that for any almost complex structure $J$ compatible with the symplectic form, and any two distinct points $(P, Q)$, there is a unique $J$-holomorphic line through $P$ and $Q$ [Gro85]. Similarly, for any almost complex structure $J$ compatible with the symplectic form, and any point and tangent vector $(P, T)$, there is a unique $J$-holomorphic line through $P$ with tangent vector $T$. This is shown in the proof of [Wen10, Theorem 6.1] (see also [McD91]).

*Outline.* Before diving further into the proof, we provide an outline of the arguments that follow. For the proofs of surjectivity and injectivity, we augment a realization $C$ of $C_0$ with an additional line to a realization of $C_1$. For injectivity, (the harder direction), we augment a 1-parameter family of realizations $C_t$ of $C_0$ with lines $L_t$ to form a family of realizations of $C_1$. We obtain these augmented configurations in three steps (performing these three steps first in the case of discrete realizations, and then in the 1-parametric version). We summarize the three steps here (stated for a single realization $C$ of $C_0$). In the first step, choose an almost complex structure $J$ which makes $C$ $J$-holomorphic, and use the results from [Gro85, McD91, Wen10] to find the unique $J$-holomorphic line $L$ passing through two points $P, Q$, or through a single point $P$ with tangency $T$, so that $L$ intersects $C$ in the two singular points or one tangency on $C_0$ required by the hypotheses. It may seem like at this point we are done, but in fact $C \cup L$ may not yet realize $C_1$ because $L$ may intersect $C$ more degenerately than required. We address this issue in the second and third steps. In step two, case (i), we adjust the line $L$ in a $C^1$ small manner locally near the points $P$ and $Q$ so that it intersects $C$ transversally at those points (in case it was originally accidentally tangent at those points). This step is unnecessary in case (ii) because the multiplicity of the tangency is fixed by the hypotheses. For either case, we perform step three, where we look at all other intersections of $L$ with $C$ outside of $P$ (and $Q$), and adjust $L$ keeping it fixed in small neighborhoods of $P$ (and $Q$) so that at the end, away from $P$ and $Q$, $C$ and $L$ only intersect in generic transverse double points. This provides the augmentation to a realization of $C_1$. Now we provide the details of this argument and apply it to prove each direction of the statement.

*Surjectivity.* We begin with the easier direction: that for each symplectic isotopy class $[C] \in \text{Iso}(C_0)$, there exists a symplectic isotopy class $[C'] \in \text{Iso}(C_1)$ such that $\Psi([C']) = [C]$.

**Step 1.** Suppose $C$ is a symplectic realization of $C_0$. Let $J$ be an almost complex structure making $C$ $J$-holomorphic. In case (i), let $P$ and $Q$ be the special points in $C$ which $L$ is required to pass through to form the configuration $C_1$ (if there are less than two singular points, one or both of these points can be chosen generically). Similarly, in case (ii), let $(P, T)$ be the point and tangent direction at which $L$ must be placed to form the configuration $C_1$ (again, the point can be chosen generically if it is not a singular point of $C_0$). Now using the results above from [Gro85, McD91, Wen10], let $L$ be the unique $J$-holomorphic line through $P$ and $Q$ or through $P$ tangent to $T$. It is possible that at this stage, $L$ passes through additional singular or tangency points of $C$ which is not desirable for the configuration $C_1$. It also is possible that the intersections at $P$ or $Q$ fail to be transverse in case (i). In case (ii), the multiplicity of the tangency at $P$ with the components of $i$ are fixed by the hypotheses so this latter problem is not relevant. We deal with the latter problem in case (i) first, and then deal with degenerate intersections away from $P$ and $Q$ in both cases.
between them and dropping the realization of the line $L$ from each $C'_t$ yields a symplectic isotopy between the realizations $C_1$ and $C_2$ of $C_0$. We show that the map $\Psi$ is surjective and injective to obtain the stated result.

Recall that Gromov proved that for any almost complex structure $J$ compatible with the symplectic form, and any two distinct points $(P, Q)$, there is a unique $J$-holomorphic line through $P$ and $Q$ [Gro85]. Similarly, for any almost complex structure $J$ compatible with the symplectic form, and any point and tangent vector $(P, T)$, there is a unique $J$-holomorphic line through $P$ with tangent vector $T$. This is shown in the proof of [Wen10, Theorem 6.1] (see also [McD91]).

*Outline.* Before diving further into the proof, we provide an outline of the arguments that follow. For the proofs of surjectivity and injectivity, we augment a realization $C$ of $C_0$ with an additional line to a realization of $C_1$. For injectivity, (the harder direction), we augment a 1-parameter family of realizations $C_t$ of $C_0$ with lines $L_t$ to form a family of realizations of $C_1$. We obtain these augmented configurations in three steps (performing these three steps first in the case of discrete realizations, and then in the 1-parametric version). We summarize the three steps here (stated for a single realization $C$ of $C_0$). In the first step, choose an almost complex structure $J$ which makes $C$ $J$-holomorphic, and use the results from [Gro85, McD91, Wen10] to find the unique $J$-holomorphic line $L$ passing through two points $P, Q$, or through a single point $P$ with tangency $T$, so that $L$ intersects $C$ in the two singular points or one tangency on $C_0$ required by the hypotheses. It may seem like at this point we are done, but in fact $C \cup L$ may not yet realize $C_1$ because $L$ may intersect $C$ more degenerately than required. We address this issue in the second and third steps. In step two, case (i), we adjust the line $L$ in a $C^1$ small manner locally near the points $P$ and $Q$ so that it intersects $C$ transversally at those points (in case it was originally accidentally tangent at those points). This step is unnecessary in case (ii) because the multiplicity of the tangency is fixed by the hypotheses. For either case, we perform step three, where we look at all other intersections of $L$ with $C$ outside of $P$ (and $Q$), and adjust $L$ keeping it fixed in small neighborhoods of $P$ (and $Q$) so that at the end, away from $P$ and $Q$, $C$ and $L$ only intersect in generic transverse double points. This provides the augmentation to a realization of $C_1$. Now we provide the details of this argument and apply it to prove each direction of the statement.

*Surjectivity.* We begin with the easier direction: that for each symplectic isotopy class $[C] \in \text{Iso}(C_0)$, there exists a symplectic isotopy class $[C'] \in \text{Iso}(C_1)$ such that $\Psi([C']) = [C]$.

*Step 1.* Suppose $C$ is a symplectic realization of $C_0$. Let $J$ be an almost complex structure making $C$ $J$-holomorphic. In case (i), let $P$ and $Q$ be the special points in $C$ which $L$ is required to pass through to form the configuration $C_1$ (if there are less than two singular points, one or both of these points can be chosen generically). Similarly, in case (ii), let $(P, T)$ be the point and tangent direction at which $L$ must be placed to form the configuration $C_1$ (again, the point can be chosen generically if it is not a singular point of $C_0$). Now using the results above from [Gro85, McD91, Wen10], let $L$ be the unique $J$-holomorphic line through $P$ and $Q$ or through $P$ tangent to $T$. It is possible that at this stage, $L$ passes through additional singular or tangency points of $C$ which is not desirable for the configuration $C_1$. It also is possible that the intersections at $P$ or $Q$ fail to be transverse in case (i). In case (ii), the multiplicity of the tangency at $P$ with the components of $i$ are fixed by the hypotheses so this latter problem is not relevant. We deal with the latter problem in case (i) first, and then deal with degenerate intersections away from $P$ and $Q$ in both cases.
M. GOLLA AND L. STARKSTON

Step 2. For case (i), we now explain how to make any required adjustments to ensure that the intersections of $L$ with $C$ at $P$ and $Q$ are transverse. Let $\tilde{U}_P$ and $\tilde{U}_Q$ be open neighborhoods of $P$ and $Q$ with disjoint closures, such that $P$ and $Q$ are the only singular points of $L \cup C$ in the closures of $\tilde{U}_P$ and $\tilde{U}_Q$. Let $U_P$ and $U_Q$ be open neighborhoods of $P$ and $Q$ such that the closures satisfy $\bar{U}_P \subset \tilde{U}_P$ and $\bar{U}_Q \subset \tilde{U}_Q$. Let $M_P$ be the moduli space of $J$-holomorphic lines which pass through $P$, and $M_Q$ the corresponding moduli space of lines through $Q$. Each of these moduli spaces is diffeomorphic to $\mathbb{CP}^1$. The subset of $M_P$ of lines which are tangent to $C$ at $P$ is a compact zero-dimensional subset (similarly for $Q$). Because this subset has codimension 2, there exists a $J$-holomorphic line $L_P$ (respectively, $L_Q$) which is $C^\infty$-close to $L$ and intersects $C$ transversally at $P$ (respectively, $Q$). Because $L_P$ and $L_Q$ are $C^\infty$-close to $L$, and the closures of the neighborhoods $\tilde{U}_P$ and $\tilde{U}_Q$ are disjoint, these lines can be spliced together symplectically. Our splicing will be a symplectic line which agrees with $L_P$ inside $U_P$, agrees with $L_Q$ inside $U_Q$, and agrees with the original $L$ outside of $\tilde{U}_P \cup \tilde{U}_Q$. Because $L$ has no intersections with $C$ in $\tilde{U}_P \setminus U_P$ or $\tilde{U}_Q \setminus U_Q$, by choosing the appropriate $C^\infty$-closeness for $L_P$ and $L_Q$ in terms of the fixed neighborhoods, we can ensure that the spliced line has no intersections with $C$ in $\tilde{U}_P \setminus U_P$ or $\tilde{U}_Q \setminus U_Q$. Therefore, any intersections of the spliced line with $C$ occur in regions where it is $J$-holomorphic, thus creating singularities of $C \cup L$ with allowable models for singular symplectic curves (in particular, the intersections are positive). Abusing notation, we rename this spliced line as $L$. Note that $C$ has not been changed.

Now the new $L \cup C$ satisfies the singularity requirements of $C_1$ near $P$ and $Q$ (in case (i) via the modification and in case (2) by the assumption constraining its intersections). However, it is still possible that $L$ passes through additional singular or tangency points of $C$ which cause it to differ from the configuration $C_1$, because it should be intersecting $C$ generically away from $P$ and $Q$.

Step 3. To make the final adjustment to $L$ so that $C \cup L$ will represent the configuration $C_1$, let $U$, $\tilde{U}$ be open sets with $\bar{U} \subset \tilde{U}$ such that $P$ and $Q$ are not in the closure of $\tilde{U}$ and any singular point of $C \cup L$ which is not one of the designated points $P$ or $Q$ is contained in $U$. For a given $J$, the space of $J$-holomorphic lines in $\mathbb{CP}^2$ has real dimension four. The subspace of lines which intersect $C$ at a particular singular point is stratified with real co-dimension at least two, and similarly the subspace of lines which intersects $C$ tangentially is stratified with real co-dimension at least two. In particular, there exists a $J$-holomorphic line $\tilde{L}$ which is $C^\infty$-close to $L$ that intersect $C$ generically inside of $U$, and has no intersections with $C$ in $\tilde{U} \setminus U$. (Again, we use the assumption that there are no intersections of $L$ with $C$ in the closure of $\tilde{U} \setminus U$.) Although $\tilde{L}$ likely does not satisfy the required intersection properties at $P$ or $Q$, we can splice together $L$ with $\tilde{L}$ inside of $\tilde{U}$. Because each $\tilde{L}$ is chosen $C^\infty$-close to $L$, we can construct a symplectic line which agrees with $\tilde{L}$ inside of $U$, agrees with $L$ outside of $\tilde{U}$, and has no intersections with $C$ in $\tilde{U} \setminus U$. Again, we rename this spliced line as $L$. Note that because $P$ and $Q$ lie outside of the $\tilde{U}$, the new $L$ still intersects $C$ in the special points there as required.

**Conclusion.** Since $C$ has not been modified, and $L$ has been constructed so that $C \cup L$ is a symplectic realization of $C_1$, we have found that $\Psi([C \cup L]) = [C]$, as desired so $\Psi$ is surjective.

**Injectivity.** To show that $\Psi$ is injective, suppose that $\Psi(A) = \Psi(B)$ for $A, B \in \text{Iso}(C_1)$. This means that $A$ is realized by a configuration $L^0 \cup C^0$ of $C_1$ and $B$ is realized by a configuration $L^1 \cup C^1$ of $C_1$ such that $C^0$ and $C^1$ are symplectically isotopic realizations of $C_0$. Using the symplectic isotopy from $C^0$ to $C^1$ as realizations of $C_0$, we extend this to a symplectic isotopy
M. GOLLA AND L. STARKSTON

Step 2. For case (i), we now explain how to make any required adjustments to ensure that the intersections of $L$ with $C$ at $P$ and $Q$ are transverse. Let $\tilde{U}_P$ and $\tilde{U}_Q$ be open neighborhoods of $P$ and $Q$ with disjoint closures, such that $P$ and $Q$ are the only singular points of $L \cup C$ in the closures of $\tilde{U}_P$ and $\tilde{U}_Q$. Let $U_P$ and $U_Q$ be open neighborhoods of $P$ and $Q$ such that the closures satisfy $\bar{U}_P \subset \tilde{U}_P$ and $\bar{U}_Q \subset \tilde{U}_Q$. Let $M_P$ be the moduli space of $J$-holomorphic lines which pass through $P$, and $M_Q$ the corresponding moduli space of lines through $Q$. Each of these moduli spaces is diffeomorphic to $\mathbb{CP}^1$. The subset of $M_P$ of lines which are tangent to $C$ at $P$ is a compact zero-dimensional subset (similarly for $Q$). Because this subset has codimension 2, there exists a $J$-holomorphic line $L_P$ (respectively, $L_Q$) which is $C^\infty$-close to $L$ and intersects $C$ transversally at $P$ (respectively, $Q$). Because $L_P$ and $L_Q$ are $C^\infty$-close to $L$, and the closures of the neighborhoods $\tilde{U}_P$ and $\tilde{U}_Q$ are disjoint, these lines can be spliced together symplectically. Our splicing will be a symplectic line which agrees with $L_P$ inside $U_P$, agrees with $L_Q$ inside $U_Q$, and agrees with the original $L$ outside of $\tilde{U}_P \cup \tilde{U}_Q$. Because $L$ has no intersections with $C$ in $\tilde{U}_P \setminus U_P$ or $\tilde{U}_Q \setminus U_Q$, by choosing the appropriate $C^\infty$-closeness for $L_P$ and $L_Q$ in terms of the fixed neighborhoods, we can ensure that the spliced line has no intersections with $C$ in $\tilde{U}_P \setminus U_P$ or $\tilde{U}_Q \setminus U_Q$. Therefore, any intersections of the spliced line with $C$ occur in regions where it is $J$-holomorphic, thus creating singularities of $C \cup L$ with allowable models for singular symplectic curves (in particular, the intersections are positive). Abusing notation, we rename this spliced line as $L$. Note that $C$ has not been changed.

Now the new $L \cup C$ satisfies the singularity requirements of $C_1$ near $P$ and $Q$ (in case (i) via the modification and in case (2) by the assumption constraining its intersections). However, it is still possible that $L$ passes through additional singular or tangency points of $C$ which cause it to differ from the configuration $C_1$, because it should be intersecting $C$ generically away from $P$ and $Q$.

Step 3. To make the final adjustment to $L$ so that $C \cup L$ will represent the configuration $C_1$, let $U$, $\tilde{U}$ be open sets with $\bar{U} \subset \tilde{U}$ such that $P$ and $Q$ are not in the closure of $\tilde{U}$ and any singular point of $C \cup L$ which is not one of the designated points $P$ or $Q$ is contained in $U$. For a given $J$, the space of $J$-holomorphic lines in $\mathbb{CP}^2$ has real dimension four. The subspace of lines which intersect $C$ at a particular singular point is stratified with real co-dimension at least two, and similarly the subspace of lines which intersects $C$ tangentially is stratified with real co-dimension at least two. In particular, there exists a $J$-holomorphic line $\tilde{L}$ which is $C^\infty$-close to $L$ that intersect $C$ generically inside of $U$, and has no intersections with $C$ in $\tilde{U} \setminus U$. (Again, we use the assumption that there are no intersections of $L$ with $C$ in the closure of $\tilde{U} \setminus U$.) Although $\tilde{L}$ likely does not satisfy the required intersection properties at $P$ or $Q$, we can splice together $L$ with $\tilde{L}$ inside of $\tilde{U}$. Because each $\tilde{L}$ is chosen $C^\infty$-close to $L$, we can construct a symplectic line which agrees with $\tilde{L}$ inside of $U$, agrees with $L$ outside of $\tilde{U}$, and has no intersections with $C$ in $\tilde{U} \setminus U$. Again, we rename this spliced line as $L$. Note that because $P$ and $Q$ lie outside of the $\tilde{U}$, the new $L$ still intersects $C$ in the special points there as required.

**Conclusion.** Since $C$ has not been modified, and $L$ has been constructed so that $C \cup L$ is a symplectic realization of $C_1$, we have found that $\Psi([C \cup L]) = [C]$, as desired so $\Psi$ is surjective.

**Injectivity.** To show that $\Psi$ is injective, suppose that $\Psi(A) = \Psi(B)$ for $A, B \in \text{Iso}(C_1)$. This means that $A$ is realized by a configuration $L^0 \cup C^0$ of $C_1$ and $B$ is realized by a configuration $L^1 \cup C^1$ of $C_1$ such that $C^0$ and $C^1$ are symplectically isotopic realizations of $C_0$. Using the symplectic isotopy from $C^0$ to $C^1$ as realizations of $C_0$, we extend this to a symplectic isotopy
from $L^0 \cup C^0$ to $L^0 \cup C^1$ as realizations of $C_1$. Let $C^t$ denote the equisingular symplectic isotopy from $C^0$ to $C^1$.

**Step 1 (1-parametric).** By Lemma 3.4, there exists a family of compatible almost complex structures $J_t$ such that $C^t$ is $J_t$-holomorphic for $t \in [0, 1]$. As $C^i \cup L^i$ is a singular symplectic curve, the space of compatible almost complex structures $\mathcal{J}^\omega(C^i \cup L^i)$ is a non-empty, contractible subspace of $\mathcal{J}^\omega(C^i)$, which is also contractible. As $\mathcal{J}^\omega(C^i \cup L^i) \subseteq \mathcal{J}^\omega(C^i)$, there are almost complex structures $J_t$, $t \in [-1, 2]$ agreeing with the previously defined $J_t$ for $t \in [0, 1]$ such that $J_{-1} \in \mathcal{J}^\omega(C^0 \cup L^0)$, $J_2 \in \mathcal{J}^\omega(C^1 \cup L^1)$, $J_t \in \mathcal{J}^\omega(C^0)$ for $t \in [-1, 0]$ and $J_t \in \mathcal{J}^\omega(C^1)$ for $t \in [1, 2]$.

Let $C^t = C^0$ for $t \in [-1, 0]$, and $C^t = C^1$ for $t \in [1, 2]$. Now $C^t$ for $t \in [-1, 2]$ is an equisingular symplectic isotopy which is $J_t$-holomorphic. Define $L^{-1} := L^0$ and $L^2 := L^1$. We construct an equisingular symplectic isotopy connecting $L^{-1} \cup C^{-1} = L^0 \cup C^0$ to $L^2 \cup C^2 = L^1 \cup C^1$.

Let $(P^t, Q^t)$ (respectively, $(P^t, T^t)$) be the two points in $C^t$ (respectively, point and tangent direction in $C^t$) which the line $L^t$ must pass through in order to satisfy the constraints of the configuration $C^1$. Even if some of these points are chosen generically, we make sure to choose them to vary smoothly with $t$, and such that $P^i$ and $Q^i$ lie on $L^i$ for $i = -1, 2$. As in the first direction, we use Gromov's and McDuff's theorems to initially define $L^t$ for $t \in [-1, 2]$ as the unique $J_t$-holomorphic line passing through $P^t$ and $Q^t$ (respectively, passing through $P^t$ with tangent direction $T^t$). Because $L^{-1}$ and $L^2$ are the unique lines satisfying the point (respectively point and tangency) conditions, this provides a 1-parameter family of $J_t$-holomorphic symplectic lines connecting $L^{-1}$ to $L^2$.

Again we need to deal with the possibility that $L^t$ may develop non-generic intersections away from $P^t$ and $Q^t$, or that the intersections at $P^t$ or $Q^t$ may become tangencies between $L^t$ and components of $C^t$ in case (i). We perform a similar $C^\infty$-small adjustment of the line near such problematic points, but now fitting this into the 1-parameter family relative to the endpoints of the family. Let $\mathbb{I} = [-1, 2]$ denote the parameter interval.

**Step 2 (1-parametric).** We again start by ensuring the intersection behavior of $L^t$ is correct near $P^t$ and $Q^t$. (Note this stage is unnecessary in case (ii) because of the stronger hypothesis.) For this, fix open sets $\hat{U}_P^t$ and $\hat{U}_Q^t$ with disjoint closures with $P_t \in U_P^t \subset \bar{U}_P^t \subset \hat{U}_P^t$ and $Q_t \in U_Q^t \subset \bar{U}_Q^t \subset \hat{U}_Q^t$ for all $t \in \mathbb{I}$, such that $L^t \cup C^t$ has no singular points in $\hat{U}_P^t \setminus U_P^t$ or $\hat{U}_Q^t \setminus U_Q^t$ and where

$$ \hat{U}_P = \bigcup_{t \in \mathbb{I}} \hat{U}_P^t \subset \mathbb{CP}^2 \times \mathbb{I} $$

and

$$ \hat{U}_Q = \bigcup_{t \in \mathbb{I}} \hat{U}_Q^t \subset \mathbb{CP}^2 \times \mathbb{I} $$

are open neighborhoods of the paths $P_t$ and $Q_t$ in $\mathbb{CP}^2 \times \mathbb{I}$ (and similarly for the smaller neighborhoods denoted without hats). Let $M_P$ be the moduli space of pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is a $J_t$-holomorphic line in $\mathbb{CP}^2$ which passes through $P_t$. Similarly, let $M_Q$ be the moduli space of pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is a $J_t$-holomorphic line in $\mathbb{CP}^2$ which passes through $Q_t$. Both $M_P$ and $M_Q$ have natural maps $\pi_P : M_P \to \mathbb{I}$ and $\pi_Q : M_Q \to \mathbb{I}$ (sending $(K^t, t)$ to $t$), and these maps are fibrations with fiber diffeomorphic to $\mathbb{CP}^1$. The subset $B_P$ of $M_P$ of pairs $(K^t, t)$ which are tangent to $C^t$ at $P^t$ is a finite set of points in each $t$-slice, which form a section or multi-section of the projection $M_P \to \mathbb{I}$. The same statement holds for the analogous subset $B_Q \subset M_Q$. As it is required to pass through $P^t$ and $Q^t$, $L^t$ gives a section of $\pi_P$ and a section
from $L^0 \cup C^0$ to $L^0 \cup C^1$ as realizations of $C_1$. Let $C^t$ denote the equisingular symplectic isotopy from $C^0$ to $C^1$.

**Step 1 (1-parametric).** By Lemma 3.4, there exists a family of compatible almost complex structures $J_t$ such that $C^t$ is $J_t$-holomorphic for $t \in [0, 1]$. As $C^i \cup L^i$ is a singular symplectic curve, the space of compatible almost complex structures $\mathcal{J}^\omega(C^i \cup L^i)$ is a non-empty, contractible subspace of $\mathcal{J}^\omega(C^i)$, which is also contractible. As $\mathcal{J}^\omega(C^i \cup L^i) \subseteq \mathcal{J}^\omega(C^i)$, there are almost complex structures $J_t$, $t \in [-1, 2]$ agreeing with the previously defined $J_t$ for $t \in [0, 1]$ such that $J_{-1} \in \mathcal{J}^\omega(C^0 \cup L^0)$, $J_2 \in \mathcal{J}^\omega(C^1 \cup L^1)$, $J_t \in \mathcal{J}^\omega(C^0)$ for $t \in [-1, 0]$ and $J_t \in \mathcal{J}^\omega(C^1)$ for $t \in [1, 2]$.

Let $C^t = C^0$ for $t \in [-1, 0]$, and $C^t = C^1$ for $t \in [1, 2]$. Now $C^t$ for $t \in [-1, 2]$ is an equisingular symplectic isotopy which is $J_t$-holomorphic. Define $L^{-1} := L^0$ and $L^2 := L^1$. We construct an equisingular symplectic isotopy connecting $L^{-1} \cup C^{-1} = L^0 \cup C^0$ to $L^2 \cup C^2 = L^1 \cup C^1$.

Let $(P^t, Q^t)$ (respectively, $(P^t, T^t)$) be the two points in $C^t$ (respectively, point and tangent direction in $C^t$) which the line $L^t$ must pass through in order to satisfy the constraints of the configuration $C^1$. Even if some of these points are chosen generically, we make sure to choose them to vary smoothly with $t$, and such that $P^i$ and $Q^i$ lie on $L^i$ for $i = -1, 2$. As in the first direction, we use Gromov's and McDuff's theorems to initially define $L^t$ for $t \in [-1, 2]$ as the unique $J_t$-holomorphic line passing through $P^t$ and $Q^t$ (respectively, passing through $P^t$ with tangent direction $T^t$). Because $L^{-1}$ and $L^2$ are the unique lines satisfying the point (respectively point and tangency) conditions, this provides a 1-parameter family of $J_t$-holomorphic symplectic lines connecting $L^{-1}$ to $L^2$.

Again we need to deal with the possibility that $L^t$ may develop non-generic intersections away from $P^t$ and $Q^t$, or that the intersections at $P^t$ or $Q^t$ may become tangencies between $L^t$ and components of $C^t$ in case (i). We perform a similar $C^\infty$-small adjustment of the line near such problematic points, but now fitting this into the 1-parameter family relative to the endpoints of the family. Let $\mathbb{I} = [-1, 2]$ denote the parameter interval.

**Step 2 (1-parametric).** We again start by ensuring the intersection behavior of $L^t$ is correct near $P^t$ and $Q^t$. (Note this stage is unnecessary in case (ii) because of the stronger hypothesis.) For this, fix open sets $\hat{U}_P^t$ and $\hat{U}_Q^t$ with disjoint closures with $P_t \in U_P^t \subset \bar{U}_P^t \subset \hat{U}_P^t$ and $Q_t \in U_Q^t \subset \bar{U}_Q^t \subset \hat{U}_Q^t$ for all $t \in \mathbb{I}$, such that $L^t \cup C^t$ has no singular points in $\hat{U}_P^t \setminus U_P^t$ or $\hat{U}_Q^t \setminus U_Q^t$ and where

$$ \hat{U}_P = \bigcup_{t \in \mathbb{I}} \hat{U}_P^t \subset \mathbb{CP}^2 \times \mathbb{I} $$

and

$$ \hat{U}_Q = \bigcup_{t \in \mathbb{I}} \hat{U}_Q^t \subset \mathbb{CP}^2 \times \mathbb{I} $$

are open neighborhoods of the paths $P_t$ and $Q_t$ in $\mathbb{CP}^2 \times \mathbb{I}$ (and similarly for the smaller neighborhoods denoted without hats). Let $M_P$ be the moduli space of pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is a $J_t$-holomorphic line in $\mathbb{CP}^2$ which passes through $P_t$. Similarly, let $M_Q$ be the moduli space of pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is a $J_t$-holomorphic line in $\mathbb{CP}^2$ which passes through $Q_t$. Both $M_P$ and $M_Q$ have natural maps $\pi_P : M_P \to \mathbb{I}$ and $\pi_Q : M_Q \to \mathbb{I}$ (sending $(K^t, t)$ to $t$), and these maps are fibrations with fiber diffeomorphic to $\mathbb{CP}^1$. The subset $B_P$ of $M_P$ of pairs $(K^t, t)$ which are tangent to $C^t$ at $P^t$ is a finite set of points in each $t$-slice, which form a section or multi-section of the projection $M_P \to \mathbb{I}$. The same statement holds for the analogous subset $B_Q \subset M_Q$. As it is required to pass through $P^t$ and $Q^t$, $L^t$ gives a section of $\pi_P$ and a section
of $\pi_Q$. As $C^t \cup L^t$ realizes the configuration $C_1$ for $t$ near $\partial\mathbb{I}$, $L^t$ avoids the bad subsets $B_P$ (respectively, $B_Q$) of $M_P$ (respectively, $M_Q$) near $\pi_P^{-1}(\partial\mathbb{I})$ (respectively, $\pi_Q^{-1}(\partial\mathbb{I})$). For $X \in \{P, Q\}$, let $L_X^t$ be a section of $\pi_X: M_X \to \mathbb{I}$ which agrees with $L^t$ near $\pi_X^{-1}(\partial\mathbb{I})$, is $C^\infty$-close to the section $L^t$ everywhere and which avoids the bad subset $B_X$. Then $L_P^t$ and $L_Q^t$ are $J_t$-holomorphic lines which are $C^\infty$-close to $L^t$. We will splice them together to a smoothly varying $\mathbb{I}$-family of symplectic lines which agree with $L_P^t$ in $U_P^t$, with $L_Q^t$ in $U_Q^t$ and with the original $L^t$ outside of $\hat{U}_P^t \cup \hat{U}_Q^t$. To make the splicing vary smoothly with $t \in \mathbb{I}$, we choose the cut-off functions to vary continuously with $t$ by letting them be the restrictions to $t$ slices of a smooth cut-off function supported on $\hat{U}_X \setminus U_X$, for $X \in \{P, Q\}$. The splicing is trivial near $\partial\mathbb{I}$. We rename the spliced family $L^t$.

**Step 3 (1-parametric).** Once the behavior near $P^t$ and $Q^t$ is correct, we next fix any overly degenerate intersections of $L^t$ with $C^t$ away from $P^t$ and $Q^t$ in either case (i) or (ii). Let $U \subset \bar{U} \subset \hat{U} \subset \mathbb{CP}^2 \times \mathbb{I}$ be open subsets whose closures do not contain $(P^t, t)$ or $(Q^t, t)$ for any $t \in \mathbb{I}$, such that every intersection point of $L^t$ with $C^t$ which is not $P^t$ or $Q^t$ is contained in $U \cap \mathbb{CP}^2 \times \{t\}$. Note that this is possible to achieve because no other intersection points can approach $P^t$ or $Q^t$ after the above modification which ensures that the multiplicities of the intersection of $L^t$ with $C^t$ at $P^t$ and $Q^t$ remain constant.

Let $\mathbb{M}$ be the moduli space of all pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is any $J_t$-holomorphic line in $\mathbb{CP}^2$. There is a natural fibration $\pi: \mathbb{M} \to \mathbb{I}$ sending $(K^t, t)$ to $t$, whose fibers $M_t$ are diffeomorphic to $\mathbb{CP}^2$ (in particular, the fibers are four-dimensional manifolds). The subset $B_t$ of $M_t$ of $J_t$-lines $K^t$ which are tangent to $C^t$ at some point other than $P^t$ in case (ii) or pass through a singular point of $C^t$ other than $P^t$ or $Q^t$ is a stratified subspace with strata of codimension at least two in $M_t$. As these subspaces vary smoothly with $t$, the union $\mathbb{B} = \cup_{t \in \mathbb{I}} B_t$ forms a stratified subspace of $\mathbb{M}$ with strata of codimension at least two. The family $(L^t, t)$ provides a section of $\pi: \mathbb{M} \to \mathbb{I}$, which avoids $\mathbb{B}$ near its endpoints, but may intersect $\mathbb{B}$ at interior values of $t$. Let $\tilde{L}^t$ be a section which agrees with $L^t$ near its end points, is $C^\infty$-close to $L^t$ everywhere and is disjoint from $\mathbb{B}$. We splice together $L^t$ and $\tilde{L}^t$ to form a smoothly varying family of symplectic lines which agree with $\tilde{L}^t$ inside of $U$ and with $L^t$ outside of $\hat{U}$. This ensures that the intersections of $L^t$ with $C^t$ near $P^t$ and $Q^t$ remain as they should be outside of $\hat{U}$ and the other intersections of $L^t$ with $C^t$ remain inside of $U$ (by $C^\infty$-closeness) and are made generic as required for the configuration $C_1$.

*Conclusion.* Replacing $L^t$ with this splicing provides the required equisingular symplectic isotopy from $C^{-1} \cup L^{-1}$ to $C^2 \cup L^2$. As $C^{-1} \cup L^{-1} = C^0 \cup L^0$ and $C^2 \cup L^2 = C^1 \cup L^1$, this shows that any two realizations of $C_1$ are equisingularly symplectically isotopic. $\square$

We can immediately recover unique isotopy classifications for small line arrangements.

**COROLLARY 5.3.** *A symplectic line arrangement with at most six lines has a unique symplectic isotopy class.*

**COROLLARY 5.4.** *Any configuration of one symplectic conic with three symplectic lines has a unique symplectic isotopy class unless it is the configuration $G$ of Figure 14 (which we obstruct in Proposition 5.21).*

*Proof.* Gromov proved there is a unique symplectic isotopy class of a conic $Q$ in $\mathbb{CP}^2$ [Gro85]. We are adding at most three lines, and we add lines tangent to the conic first. Therefore we add the first line, $L_1$ which intersects this conic either tangentially at a point or transversally at two points and this does not change the uniqueness of the isotopy classification by Proposition 5.1.
of $\pi_Q$. As $C^t \cup L^t$ realizes the configuration $C_1$ for $t$ near $\partial\mathbb{I}$, $L^t$ avoids the bad subsets $B_P$ (respectively, $B_Q$) of $M_P$ (respectively, $M_Q$) near $\pi_P^{-1}(\partial\mathbb{I})$ (respectively, $\pi_Q^{-1}(\partial\mathbb{I})$). For $X \in \{P, Q\}$, let $L_X^t$ be a section of $\pi_X: M_X \to \mathbb{I}$ which agrees with $L^t$ near $\pi_X^{-1}(\partial\mathbb{I})$, is $C^\infty$-close to the section $L^t$ everywhere and which avoids the bad subset $B_X$. Then $L_P^t$ and $L_Q^t$ are $J_t$-holomorphic lines which are $C^\infty$-close to $L^t$. We will splice them together to a smoothly varying $\mathbb{I}$-family of symplectic lines which agree with $L_P^t$ in $U_P^t$, with $L_Q^t$ in $U_Q^t$ and with the original $L^t$ outside of $\hat{U}_P^t \cup \hat{U}_Q^t$. To make the splicing vary smoothly with $t \in \mathbb{I}$, we choose the cut-off functions to vary continuously with $t$ by letting them be the restrictions to $t$ slices of a smooth cut-off function supported on $\hat{U}_X \setminus U_X$, for $X \in \{P, Q\}$. The splicing is trivial near $\partial\mathbb{I}$. We rename the spliced family $L^t$.

**Step 3 (1-parametric).** Once the behavior near $P^t$ and $Q^t$ is correct, we next fix any overly degenerate intersections of $L^t$ with $C^t$ away from $P^t$ and $Q^t$ in either case (i) or (ii). Let $U \subset \bar{U} \subset \hat{U} \subset \mathbb{CP}^2 \times \mathbb{I}$ be open subsets whose closures do not contain $(P^t, t)$ or $(Q^t, t)$ for any $t \in \mathbb{I}$, such that every intersection point of $L^t$ with $C^t$ which is not $P^t$ or $Q^t$ is contained in $U \cap \mathbb{CP}^2 \times \{t\}$. Note that this is possible to achieve because no other intersection points can approach $P^t$ or $Q^t$ after the above modification which ensures that the multiplicities of the intersection of $L^t$ with $C^t$ at $P^t$ and $Q^t$ remain constant.

Let $\mathbb{M}$ be the moduli space of all pairs $(K^t, t)$ where $t \in \mathbb{I}$ and $K^t$ is any $J_t$-holomorphic line in $\mathbb{CP}^2$. There is a natural fibration $\pi: \mathbb{M} \to \mathbb{I}$ sending $(K^t, t)$ to $t$, whose fibers $M_t$ are diffeomorphic to $\mathbb{CP}^2$ (in particular, the fibers are four-dimensional manifolds). The subset $B_t$ of $M_t$ of $J_t$-lines $K^t$ which are tangent to $C^t$ at some point other than $P^t$ in case (ii) or pass through a singular point of $C^t$ other than $P^t$ or $Q^t$ is a stratified subspace with strata of codimension at least two in $M_t$. As these subspaces vary smoothly with $t$, the union $\mathbb{B} = \cup_{t \in \mathbb{I}} B_t$ forms a stratified subspace of $\mathbb{M}$ with strata of codimension at least two. The family $(L^t, t)$ provides a section of $\pi: \mathbb{M} \to \mathbb{I}$, which avoids $\mathbb{B}$ near its endpoints, but may intersect $\mathbb{B}$ at interior values of $t$. Let $\tilde{L}^t$ be a section which agrees with $L^t$ near its end points, is $C^\infty$-close to $L^t$ everywhere and is disjoint from $\mathbb{B}$. We splice together $L^t$ and $\tilde{L}^t$ to form a smoothly varying family of symplectic lines which agree with $\tilde{L}^t$ inside of $U$ and with $L^t$ outside of $\hat{U}$. This ensures that the intersections of $L^t$ with $C^t$ near $P^t$ and $Q^t$ remain as they should be outside of $\hat{U}$ and the other intersections of $L^t$ with $C^t$ remain inside of $U$ (by $C^\infty$-closeness) and are made generic as required for the configuration $C_1$.

**Conclusion.** Replacing $L^t$ with this splicing provides the required equisingular symplectic isotopy from $C^{-1} \cup L^{-1}$ to $C^2 \cup L^2$. As $C^{-1} \cup L^{-1} = C^0 \cup L^0$ and $C^2 \cup L^2 = C^1 \cup L^1$, this shows that any two realizations of $C_1$ are equisingularly symplectically isotopic. $\square$

We can immediately recover unique isotopy classifications for small line arrangements.

**COROLLARY 5.3.** A symplectic line arrangement with at most six lines has a unique symplectic isotopy class.

**COROLLARY 5.4.** Any configuration of one symplectic conic with three symplectic lines has a unique symplectic isotopy class unless it is the configuration $\mathcal{G}$ of Figure 14 (which we obstruct in Proposition 5.21).

**Proof.** Gromov proved there is a unique symplectic isotopy class of a conic $Q$ in $\mathbb{CP}^2$ [Gro85]. We are adding at most three lines, and we add lines tangent to the conic first. Therefore we add the first line, $L_1$ which intersects this conic either tangentially at a point or transversally at two points and this does not change the uniqueness of the isotopy classification by Proposition 5.1.
<figure><img src="image_14.png" /><figcaption>FIGURE 5. Configuration $\mathcal{G}_3$.</figcaption></figure>

If the second line $L_2$ is tangent to the conic, it must intersect $L_1$ generically off of $Q$. Similarly, if all three lines are tangent to $Q$, they must intersect generically unless they form the configuration $\mathcal{G}$ where they intersect at a triple point away from $Q$. Therefore, tangent lines can be added with no further constraint than their tangency to the conic at a point, and thus Proposition 5.1 suffices in these cases. For lines being added with no tangency condition to $Q$, the only combinatorial constraints can be that $L_i$ should pass through an intersection point of $Q \cap L_j$ or $L_j \cap L_k$ for $j, k < i$. As $L_i$ can intersect $L_j$ at most once, the combinatorics requires $L_i$ to pass through at most $i - 1 \le 2$ points, so Proposition 5.1 suffices to prove there is a unique symplectic isotopy class of such a configuration. $\square$

Our second strategy to extend the known list of unique symplectic isotopy classes of reducible curves is to use birational transformations to modify configurations to collections of curves where at most one of the components is degree greater than one. We use a birational equivalence to relate the reducible configuration of interest to a reducible configuration we can understand through Proposition 5.1. We now give a collection of examples using this technique, which we use later in the paper.

5.1.1 *Two conics with a common tangent line.* Let $\mathcal{G}_3$ denote the configuration consisting of two conics $Q_1$ and $Q_2$ and a line $L_1$ tangent to both $Q_1$ and $Q_2$ (at different points) such that $Q_1$ and $Q_2$ intersect at one point with multiplicity three and at another point transversally. See Figure 5.

**PROPOSITION 5.5.** *The configuration $\mathcal{G}_3$ has a unique equisingular symplectic isotopy class.*

Let $\mathcal{A}$ denote the augmented configuration obtained from $\mathcal{G}_3$ by adding two additional lines $L_2$ and $L_3$ where $L_2$ is tangent to $Q_1$ and $Q_2$ at their multiplicity three tangency point and $L_3$ intersects $Q_1$ transversally at its tangent point with $L_1$ and its tangent point with $Q_2$. See the upper left part of Figure 6.

Let $\mathcal{B}$ denote the configuration consisting of a conic $R$ and four lines $M_1, M_2, M_3, M_4$ such that $M_1$ and $M_2$ are tangent to $R$, the intersection of $M_3$ with $M_2$ lies on $R$, the intersection of $M_3$ with $M_4$ lies on $R$ and $M_1, M_2$ and $M_4$ intersect at a triple point. See the lower right part of Figure 6.

**LEMMA 5.6.** *There is a birational equivalence between $\mathcal{A}$ and $\mathcal{B}$.*

*Proof.* Starting with a realization of $\mathcal{A}$, blow up twice at the multiplicity three tangential intersection of $Q_1$ with $Q_2$ and once at the tangential intersection of $L_1$ with $Q_1$. The resulting configuration in $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ is shown in the upper right part of Figure 6. There are three exceptional divisors $D_1, D_2, D_3$ where $D_2$ and $D_3$ have self-intersection $-1$ and $D_1$ has self-intersection $-2$. The proper transforms of $L_2$ and $L_3$ have self-intersection $-1$ so they can be symplectically blown down. After this the proper transform of $D_1$ becomes a $(-1)$-exceptional sphere which
<figure><img src="image_14.png" /><figcaption>FIGURE 5. Configuration $\mathcal{G}_3$.</figcaption></figure>

If the second line $L_2$ is tangent to the conic, it must intersect $L_1$ generically off of $Q$. Similarly, if all three lines are tangent to $Q$, they must intersect generically unless they form the configuration $\mathcal{G}$ where they intersect at a triple point away from $Q$. Therefore, tangent lines can be added with no further constraint than their tangency to the conic at a point, and thus Proposition 5.1 suffices in these cases. For lines being added with no tangency condition to $Q$, the only combinatorial constraints can be that $L_i$ should pass through an intersection point of $Q \cap L_j$ or $L_j \cap L_k$ for $j, k < i$. As $L_i$ can intersect $L_j$ at most once, the combinatorics requires $L_i$ to pass through at most $i - 1 \le 2$ points, so Proposition 5.1 suffices to prove there is a unique symplectic isotopy class of such a configuration. $\square$

Our second strategy to extend the known list of unique symplectic isotopy classes of reducible curves is to use birational transformations to modify configurations to collections of curves where at most one of the components is degree greater than one. We use a birational equivalence to relate the reducible configuration of interest to a reducible configuration we can understand through Proposition 5.1. We now give a collection of examples using this technique, which we use later in the paper.

5.1.1 *Two conics with a common tangent line.* Let $\mathcal{G}_3$ denote the configuration consisting of two conics $Q_1$ and $Q_2$ and a line $L_1$ tangent to both $Q_1$ and $Q_2$ (at different points) such that $Q_1$ and $Q_2$ intersect at one point with multiplicity three and at another point transversally. See Figure 5.

**PROPOSITION 5.5.** *The configuration $\mathcal{G}_3$ has a unique equisingular symplectic isotopy class.*

Let $\mathcal{A}$ denote the augmented configuration obtained from $\mathcal{G}_3$ by adding two additional lines $L_2$ and $L_3$ where $L_2$ is tangent to $Q_1$ and $Q_2$ at their multiplicity three tangency point and $L_3$ intersects $Q_1$ transversally at its tangent point with $L_1$ and its tangent point with $Q_2$. See the upper left part of Figure 6.

Let $\mathcal{B}$ denote the configuration consisting of a conic $R$ and four lines $M_1, M_2, M_3, M_4$ such that $M_1$ and $M_2$ are tangent to $R$, the intersection of $M_3$ with $M_2$ lies on $R$, the intersection of $M_3$ with $M_4$ lies on $R$ and $M_1, M_2$ and $M_4$ intersect at a triple point. See the lower right part of Figure 6.

**LEMMA 5.6.** *There is a birational equivalence between $\mathcal{A}$ and $\mathcal{B}$.*

*Proof.* Starting with a realization of $\mathcal{A}$, blow up twice at the multiplicity three tangential intersection of $Q_1$ with $Q_2$ and once at the tangential intersection of $L_1$ with $Q_1$. The resulting configuration in $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ is shown in the upper right part of Figure 6. There are three exceptional divisors $D_1, D_2, D_3$ where $D_2$ and $D_3$ have self-intersection $-1$ and $D_1$ has self-intersection $-2$. The proper transforms of $L_2$ and $L_3$ have self-intersection $-1$ so they can be symplectically blown down. After this the proper transform of $D_1$ becomes a $(-1)$-exceptional sphere which
<figure><img src="image_15.png" /><figcaption>FIGURE 6. Birational equivalence from $\mathcal{A}$ to $\mathcal{B}$. The upper right and lower left configurations are the same, just redrawn indicating a symplectomorphism of $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ identifying the vertical +1-curve with $\mathbb{CP}^1$.</figcaption></figure>

can be blown down as well. The resulting curve has configuration type $\mathcal{B}$ as in the bottom row of Figure 6. $\square$

*Proof of Proposition 5.5.* As $\mathcal{A}$ is obtained from $\mathcal{G}_3$ by adding one line with a point tangency condition, and one line with two transverse intersection conditions, $\mathcal{G}_3$ has a unique equisin- gular symplectic isotopy class if and only if $\mathcal{A}$ does by Proposition 5.1. By Lemma 5.6, and Corollary 3.19, $\mathcal{A}$ has a unique equisingular symplectic isotopy class if and only if $\mathcal{B}$ does. We prove $\mathcal{B}$ has a unique symplectic isotopy class by iteratively applying Proposition 5.1 to add lines to the configuration of a single conic $R$ which has a unique symplectic isotopy class by Gromov [Gro85]. We apply Proposition 5.1 four times to add the four lines, starting with the two tangent lines $M_1$ and $M_2$, then adding $M_3$ transversally through the tangent intersection of $M_2$ with $R$, and finally adding $M_4$ through $M_1 \cap M_2$ and the other intersection of $M_3$ with $R$. Therefore, $\mathcal{B}$ has a unique equisingular symplectic isotopy class so $\mathcal{G}_3$ does as well. $\square$

*Remark 5.7.* The configurations $\mathcal{A}$ and $\mathcal{B}$ were not pulled out of thin air, but rather come from augmenting a birational derivation obtained using Theorem 3.6 and Lemma 3.5. To find these configurations, start with $\mathcal{G}_3$ and blow up twice at the tangency of $Q_1$ and $Q_2$ and once at the tangency of $Q_1$ and $L_1$ to make these intersections transverse and to bring the self-intersection number of $Q_1$ to +1. Keep track of the exceptional divisors in the total transform. Now apply Theorem 3.6 to identify the proper transform of $Q_1$ with $\mathbb{CP}^1$ and determine the possible homol- ogy classes of the other curves in the configuration in terms of the standard basis (they are uniquely determined up to re-indexing). Next use Lemma 3.5 to blow down exceptional curves in classes $e_1, e_2, e_3$ and observe the effect on the configuration (it will descend to $\mathcal{B}$). To change this from a birational derivation to a birational equivalence, we need to add in curves represent- ing the $e_i$ which are not represented by curves in the configuration. Back-tracking these curves to $\mathcal{G}_3$ we find the two lines we need to augment $\mathcal{G}_3$ by to obtain $\mathcal{A}$.
<figure><img src="image_15.png" /><figcaption>FIGURE 6. Birational equivalence from $\mathcal{A}$ to $\mathcal{B}$. The upper right and lower left configurations are the same, just redrawn indicating a symplectomorphism of $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ identifying the vertical +1-curve with $\mathbb{CP}^1$.</figcaption></figure>

can be blown down as well. The resulting curve has configuration type $\mathcal{B}$ as in the bottom row of Figure 6. $\square$

*Proof of Proposition 5.5.* As $\mathcal{A}$ is obtained from $\mathcal{G}_3$ by adding one line with a point tangency condition, and one line with two transverse intersection conditions, $\mathcal{G}_3$ has a unique equisingular symplectic isotopy class if and only if $\mathcal{A}$ does by Proposition 5.1. By Lemma 5.6, and Corollary 3.19, $\mathcal{A}$ has a unique equisingular symplectic isotopy class if and only if $\mathcal{B}$ does. We prove $\mathcal{B}$ has a unique symplectic isotopy class by iteratively applying Proposition 5.1 to add lines to the configuration of a single conic $R$ which has a unique symplectic isotopy class by Gromov [Gro85]. We apply Proposition 5.1 four times to add the four lines, starting with the two tangent lines $M_1$ and $M_2$, then adding $M_3$ transversally through the tangent intersection of $M_2$ with $R$, and finally adding $M_4$ through $M_1 \cap M_2$ and the other intersection of $M_3$ with $R$. Therefore, $\mathcal{B}$ has a unique equisingular symplectic isotopy class so $\mathcal{G}_3$ does as well. $\square$

*Remark 5.7.* The configurations $\mathcal{A}$ and $\mathcal{B}$ were not pulled out of thin air, but rather come from augmenting a birational derivation obtained using Theorem 3.6 and Lemma 3.5. To find these configurations, start with $\mathcal{G}_3$ and blow up twice at the tangency of $Q_1$ and $Q_2$ and once at the tangency of $Q_1$ and $L_1$ to make these intersections transverse and to bring the self-intersection number of $Q_1$ to +1. Keep track of the exceptional divisors in the total transform. Now apply Theorem 3.6 to identify the proper transform of $Q_1$ with $\mathbb{CP}^1$ and determine the possible homology classes of the other curves in the configuration in terms of the standard basis (they are uniquely determined up to re-indexing). Next use Lemma 3.5 to blow down exceptional curves in classes $e_1, e_2, e_3$ and observe the effect on the configuration (it will descend to $\mathcal{B}$). To change this from a birational derivation to a birational equivalence, we need to add in curves representing the $e_i$ which are not represented by curves in the configuration. Back-tracking these curves to $\mathcal{G}_3$ we find the two lines we need to augment $\mathcal{G}_3$ by to obtain $\mathcal{A}$.
<figure><img src="image_16.png" /><figcaption>FIGURE 7. Birational transformation between $\mathcal{A}_2$ (left) and $\mathcal{B}_2$ (right).</figcaption></figure>

We can prove similar uniqueness statements for configurations of two conics with a common tangent line when the conics intersect more generically. Let $\mathcal{G}_2$ denote the configuration of two conics simply tangent to each other at one point with a line tangent to both conics away from their intersections. Let $\mathcal{G}_1$ denote two transversally intersecting conics with a line tangent to both.

**PROPOSITION 5.8.** The configuration $\mathcal{G}_2$ has a unique equisingular symplectic isotopy class.

*Proof.* Let the conic components of $\mathcal{G}_2$ be denoted by $Q_1, Q_2$ and the line by $L_1$. By Proposition 5.1, the symplectic isotopy classification of $\mathcal{G}_2$ is equivalent to the classification for the augmented configuration $\mathcal{A}_2$ obtained from $\mathcal{G}_2$ by adding two more lines $L_2$ through the tangential intersection of $Q_1$ with $Q_2$ and the tangency point $L \cap Q_2$. There is a birational equivalence of $\mathcal{A}_2$ to a configuration $\mathcal{B}_2$ consisting of a single conic, two tangent lines, a line passing through one of the tangency points (and, otherwise, generic), and a line passing through the intersection of the two tangent lines (and, otherwise, generic). (See Figure 7.) Here $\mathcal{B}_2$ can be built from the single conic configuration by repeated applications of Proposition 5.1 so it has a unique equisingular symplectic isotopy class. $\square$

**PROPOSITION 5.9.** The configuration $\mathcal{G}_1$ has a unique equisingular symplectic isotopy class.

*Proof.* Let the conic components of $\mathcal{G}_1$ be denoted by $Q_1, Q_2$ and the line by $L_1$. Fix two of the four intersection points $q_1, q_2$ of $Q_1$ with $Q_2$ and let $p = L_1 \cap Q_2$. By Proposition 5.1, the symplectic isotopy classification of $\mathcal{G}_1$ is equivalent to the classification for the augmented configuration $\mathcal{A}_1$ obtained from $\mathcal{G}_1$ by adding three lines passing transversally through the three pairs of the points $p, q_1, q_2$. Here $\mathcal{A}_1$ is birationally equivalent to $\mathcal{B}_1$, the configuration built from one conic with one tangent line, and four other lines intersecting as in Figure 8. The configuration $\mathcal{B}_1$ can be built from the single conic configuration by repeated applications of Proposition 5.1 so it has a unique equisingular symplectic isotopy class. $\square$

By a similar set of birational transformations, we can prove uniqueness of the symplectic isotopy class of two conics without the additional tangent line. There are five ways that two conics can intersect each other: transversally at four points, with one simple tangency and two transverse points, with one triple tangency and one transverse point, with two simple tangencies, or with one quadruple tangency. Note that the last two configurations cannot be realized with an additional common tangent line (see Proposition 5.23).

**PROPOSITION 5.10.** Any of the five configurations of two positively intersecting symplectic conics has a unique equisingular symplectic isotopy class.
<figure><img src="image_16.png" /><figcaption>FIGURE 7. Birational transformation between $\mathcal{A}_2$ (left) and $\mathcal{B}_2$ (right).</figcaption></figure>

We can prove similar uniqueness statements for configurations of two conics with a common tangent line when the conics intersect more generically. Let $\mathcal{G}_2$ denote the configuration of two conics simply tangent to each other at one point with a line tangent to both conics away from their intersections. Let $\mathcal{G}_1$ denote two transversally intersecting conics with a line tangent to both.

**PROPOSITION 5.8.** *The configuration $\mathcal{G}_2$ has a unique equisingular symplectic isotopy class.*

*Proof.* Let the conic components of $\mathcal{G}_2$ be denoted by $Q_1, Q_2$ and the line by $L_1$. By Proposition 5.1, the symplectic isotopy classification of $\mathcal{G}_2$ is equivalent to the classification for the augmented configuration $\mathcal{A}_2$ obtained from $\mathcal{G}_2$ by adding two more lines $L_2$ through the tangential intersection of $Q_1$ with $Q_2$ and the tangency point $L \cap Q_2$. There is a birational equivalence of $\mathcal{A}_2$ to a configuration $\mathcal{B}_2$ consisting of a single conic, two tangent lines, a line passing through one of the tangency points (and, otherwise, generic), and a line passing through the intersection of the two tangent lines (and, otherwise, generic). (See Figure 7.) Here $\mathcal{B}_2$ can be built from the single conic configuration by repeated applications of Proposition 5.1 so it has a unique equisingular symplectic isotopy class. $\square$

**PROPOSITION 5.9.** *The configuration $\mathcal{G}_1$ has a unique equisingular symplectic isotopy class.*

*Proof.* Let the conic components of $\mathcal{G}_1$ be denoted by $Q_1, Q_2$ and the line by $L_1$. Fix two of the four intersection points $q_1, q_2$ of $Q_1$ with $Q_2$ and let $p = L_1 \cap Q_2$. By Proposition 5.1, the symplectic isotopy classification of $\mathcal{G}_1$ is equivalent to the classification for the augmented configuration $\mathcal{A}_1$ obtained from $\mathcal{G}_1$ by adding three lines passing transversally through the three pairs of the points $p, q_1, q_2$. Here $\mathcal{A}_1$ is birationally equivalent to $\mathcal{B}_1$, the configuration built from one conic with one tangent line, and four other lines intersecting as in Figure 8. The configuration $\mathcal{B}_1$ can be built from the single conic configuration by repeated applications of Proposition 5.1 so it has a unique equisingular symplectic isotopy class. $\square$

By a similar set of birational transformations, we can prove uniqueness of the symplectic isotopy class of two conics without the additional tangent line. There are five ways that two conics can intersect each other: transversally at four points, with one simple tangency and two transverse points, with one triple tangency and one transverse point, with two simple tangencies, or with one quadruple tangency. Note that the last two configurations cannot be realized with an additional common tangent line (see Proposition 5.23).

**PROPOSITION 5.10.** *Any of the five configurations of two positively intersecting symplectic conics has a unique equisingular symplectic isotopy class.*
<figure><img src="image_17.png" /><figcaption>FIGURE 8. Birational transformation between $A_1$ (left) and $B_1$ (right).</figcaption></figure>

*Proof.* Blow up the configuration three times at intersection points between the two conics. This transforms each of the conics to a $+1$-sphere, and includes three exceptional divisors (whose intersection configuration depends on the configuration of conics we started with). Regardless of the starting configuration, the pairwise intersections between curves in the configuration (including the exceptional divisors and proper transforms of the conics) are transverse and each pair of curves intersects at most once. By McDuff's Theorem 3.6, there is a symplectomorphism of the resulting $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ which identifies the proper transform of one of the conics with $\mathbb{CP}^1$. By Lemma 3.7, the proper transform of the other conic represents the class $h = [\mathbb{CP}^1]$, and the exceptional divisors represent classes of the form $h - e_i - e_j$ or $e_i - e_j$. By Lemma 3.5, we can realize new exceptional spheres in the classes $e_i$ which intersect the configuration positively, and blow these down. Afterwards, we can blow down any exceptional spheres which then appears in the configuration representing a class $e_i$. Repeating this if needed, we reach a configuration of at most five curves in $\mathbb{CP}^2$, each representing the class $h = [\mathbb{CP}^1]$. Therefore, a configuration of at most five symplectic lines can be birationally derived from the original configuration of two conics. As any configuration of at most five lines has a unique non-empty symplectic isotopy class by Proposition 5.1, we conclude that the configuration of two conics has a unique non-empty symplectic isotopy class by Proposition 3.18. $\square$

COROLLARY 5.11. Any configuration of two conics with one line has a unique equisingular symplectic isotopy class (which is empty only if it is obstructed in Proposition 5.23).

*Proof.* This follows from Propositions 5.10 and 5.1 when the line is not required to be tangent to both conics. This is because the line must intersect each conic exactly twice (with multiplicity), and the only special points occur at the intersection of both conics (so a tangent line could not go through any special points, and a transverse line could go through at most two special points). When the line is tangent to both conics, this follows from Propositions 5.5, 5.8, 5.9, and 5.23. $\square$

5.1.2 *Unisingular curves with a maximally tangent line.* Suppose $(C, 0)$ is the germ of a (not necessarily locally irreducible) singularity with multiplicity sequence $[d-1]$. Up to topological equivalence, $(C, 0)$ is determined by a partition $d = (d_1, \dots, d_m)$ of $d-1$, i.e. an un-ordered $m$-tuple of positive integers $d_1, \dots, d_m$ such that $d_1 + \dots + d_m = d-1$; here $m$ is the number of branches of $(C, 0)$. Therefore, if $(C, 0)$ has multiplicity sequence $[d-1]$, we say that it is of type $d$ if the corresponding partition is $d$. Note that link of a singularity of type $d$ is obtained by cabling the link given by $m$ fibers of the Hopf fibration with cabling parameters $(d_1, d_1+1), \dots, (d_m, d_m+1)$.
<figure><img src="image_17.png" /><figcaption>FIGURE 8. Birational transformation between $A_1$ (left) and $B_1$ (right).</figcaption></figure>

*Proof.* Blow up the configuration three times at intersection points between the two conics. This transforms each of the conics to a +1-sphere, and includes three exceptional divisors (whose intersection configuration depends on the configuration of conics we started with). Regardless of the starting configuration, the pairwise intersections between curves in the configuration (including the exceptional divisors and proper transforms of the conics) are transverse and each pair of curves intersects at most once. By McDuff's Theorem 3.6, there is a symplectomorphism of the resulting $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$ which identifies the proper transform of one of the conics with $\mathbb{CP}^1$. By Lemma 3.7, the proper transform of the other conic represents the class $h = [\mathbb{CP}^1]$, and the exceptional divisors represent classes of the form $h - e_i - e_j$ or $e_i - e_j$. By Lemma 3.5, we can realize new exceptional spheres in the classes $e_i$ which intersect the configuration positively, and blow these down. Afterwards, we can blow down any exceptional spheres which then appears in the configuration representing a class $e_i$. Repeating this if needed, we reach a configuration of at most five curves in $\mathbb{CP}^2$, each representing the class $h = [\mathbb{CP}^1]$. Therefore, a configuration of at most five symplectic lines can be birationally derived from the original configuration of two conics. As any configuration of at most five lines has a unique non-empty symplectic isotopy class by Proposition 5.1, we conclude that the configuration of two conics has a unique non-empty symplectic isotopy class by Proposition 3.18. $\square$

COROLLARY 5.11. Any configuration of two conics with one line has a unique equisingular symplectic isotopy class (which is empty only if it is obstructed in Proposition 5.23).

*Proof.* This follows from Propositions 5.10 and 5.1 when the line is not required to be tangent to both conics. This is because the line must intersect each conic exactly twice (with multiplicity), and the only special points occur at the intersection of both conics (so a tangent line could not go through any special points, and a transverse line could go through at most two special points). When the line is tangent to both conics, this follows from Propositions 5.5, 5.8, 5.9, and 5.23. $\square$

5.1.2 *Unisingular curves with a maximally tangent line.* Suppose $(C, 0)$ is the germ of a (not necessarily locally irreducible) singularity with multiplicity sequence $[d-1]$. Up to topological equivalence, $(C, 0)$ is determined by a partition $d = (d_1, \dots, d_m)$ of $d-1$, i.e. an un-ordered $m$-tuple of positive integers $d_1, \dots, d_m$ such that $d_1 + \dots + d_m = d-1$; here $m$ is the number of branches of $(C, 0)$. Therefore, if $(C, 0)$ has multiplicity sequence $[d-1]$, we say that it is of type $d$ if the corresponding partition is $d$. Note that link of a singularity of type $d$ is obtained by cabling the link given by $m$ fibers of the Hopf fibration with cabling parameters $(d_1, d_1+1), \dots, (d_m, d_m+1)$.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

PROPOSITION 5.12. Let $d$ be a positive integer and $\mathbf{d}$ a partition of $d-1$. Let $\mathcal{C}$ denote the configuration with the following two irreducible components:

(i) $\mathcal{C}$ is a rational curve of degree $d$ in with a singular point $P$ with multiplicity sequence $[d-1]$ and of type $\mathbf{d}$;

(ii) $L$ is a line with a tangency of order $d$ with $\mathcal{C}$ at a point $Q$, where $P \neq Q$.

Then $\mathcal{C}$ has a unique equisingular symplectic isotopy class, and this unique isotopy class contains a complex curve.

*Proof.* Suppose that $\mathcal{C}$ has $m$ branches at $P$; number them so that each of them has a singularity of multiplicity $d_i$, where $\mathbf{d} = [d_1, \dots, d_m]$ is the given partition of $d-1$. In particular, $d_1 + \dots + d_m = d-1$.

By iteratively applying Proposition 5.1, the uniqueness of the symplectic isotopy class of the configuration $\mathcal{C}$ is equivalent to the uniqueness of the symplectic isotopy class of a configuration $\mathcal{C}'$ obtained from $\mathcal{C}$ by adding $m$ distinct lines tangent to $\mathcal{C}$ at $P$ along the $m$ distinct branches, together with one line that passes transversally through $P$ and $Q$. Note that for degree reasons the only other intersections are transverse double points between $L$ and each of the $m$ tangent lines. Note that for the tangent lines, to satisfy the hypotheses of Proposition 5.1, it is important that they have no other intersections with $\mathcal{C}$. This is because the intersection at $P$ has multiplicity $d$: a generic line through $P$ has intersection multiplicities $d_1, \dots, d_m$ with its branches; if $L$ is tangent to the $i$th branch of $\mathcal{C}$ at $P$, then its intersection with $\mathcal{C}$ at $P$ satisfies:

$$ (C \cdot L)_P \geq d_1 + \dots + d_{i-1} + (d_i + 1) + d_{i+1} + \dots + d_m = d, $$

which simultaneously implies that $(C \cdot L)_P = d$ and that the intersection multiplicity of $L$ with the $i$th branch is exactly $d_i + 1$.

By Lemma 5.13, there is a birational equivalence from $\mathcal{C}'$ to a configuration of $m+3$ lines with a single $(m+1)$-fold point and a single triple point. This latter configuration has a unique symplectic isotopy class by Proposition 5.1. Therefore by Corollary 3.19, the configurations $\mathcal{C}'$ has a unique symplectic isotopy class containing complex curve representatives, and thus $\mathcal{C}$ does as well. $\square$

LEMMA 5.13. Let $\mathcal{C}'$ denote the configuration above, obtained from $\mathcal{C}$ by adding $m$ lines tangent to $\mathcal{C}$ at $P$ as well as one line through $P$ and $Q$. Then the configuration $\mathcal{C}'$ is birationally equivalent to a configuration of $m+3$ lines where $m+1$ intersect at a single point and the other two intersect the $m+1$st at a triple point. (See Figure 9.)

*Proof.* Blowing up at the singular point of $\mathcal{C}$ yields a smooth rational curve, the proper transform of $\mathcal{C}$, with $m$ points of tangency with the exceptional divisor $E_0$, of order $d_1, \dots, d_m$ respectively. Blow up $d_i$ times at the $i$th tangency, so that the proper transform of $E_0$ is disjoint from that of $\mathcal{C}$. Next, blow up $d-1$ times at the tangency of $\mathcal{C}$ with $L$, so that their proper transforms intersect transversely.

We obtain the configuration of rational curves represented in Figure 10.

The proper transform of $\mathcal{C}$ has self-intersection $+1 = d^2 - (d-1)^2 - 2(d-1)$, and is a smooth sphere. We identify it with $\mathbb{CP}^1$ using Theorem 3.6 and use this to determine the homology classes of the other curves in the configuration.

By Lemma 3.7, and their pairwise disjointness, the vertical $(-1)$-curves intersecting $\mathcal{C}$ must represent classes $h - e_0 - e_1^i$ for $i = 1, \dots, m$ with the final vertical $(-1)$-curve which also intersects $L$ in the class $h - e_0 - e_1^0$. The chains of $(-2)$-curves emanating from these $(-1)$-curves
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

PROPOSITION 5.12. Let $d$ be a positive integer and $\mathbf{d}$ a partition of $d-1$. Let $\mathcal{C}$ denote the configuration with the following two irreducible components:

(i) $\mathcal{C}$ is a rational curve of degree $d$ in with a singular point $P$ with multiplicity sequence $[d-1]$ and of type $\mathbf{d}$;

(ii) $L$ is a line with a tangency of order $d$ with $\mathcal{C}$ at a point $Q$, where $P \neq Q$.

Then $\mathcal{C}$ has a unique equisingular symplectic isotopy class, and this unique isotopy class contains a complex curve.

*Proof.* Suppose that $\mathcal{C}$ has $m$ branches at $P$; number them so that each of them has a singularity of multiplicity $d_i$, where $\mathbf{d} = [d_1, \dots, d_m]$ is the given partition of $d-1$. In particular, $d_1 + \dots + d_m = d-1$.

By iteratively applying Proposition 5.1, the uniqueness of the symplectic isotopy class of the configuration $\mathcal{C}$ is equivalent to the uniqueness of the symplectic isotopy class of a configuration $\mathcal{C}'$ obtained from $\mathcal{C}$ by adding $m$ distinct lines tangent to $\mathcal{C}$ at $P$ along the $m$ distinct branches, together with one line that passes transversally through $P$ and $Q$. Note that for degree reasons the only other intersections are transverse double points between $L$ and each of the $m$ tangent lines. Note that for the tangent lines, to satisfy the hypotheses of Proposition 5.1, it is important that they have no other intersections with $\mathcal{C}$. This is because the intersection at $P$ has multiplicity $d$: a generic line through $P$ has intersection multiplicities $d_1, \dots, d_m$ with its branches; if $L$ is tangent to the $i$th branch of $\mathcal{C}$ at $P$, then its intersection with $\mathcal{C}$ at $P$ satisfies:

$$ (C \cdot L)_P \geq d_1 + \dots + d_{i-1} + (d_i + 1) + d_{i+1} + \dots + d_m = d, $$

which simultaneously implies that $(C \cdot L)_P = d$ and that the intersection multiplicity of $L$ with the $i$th branch is exactly $d_i + 1$.

By Lemma 5.13, there is a birational equivalence from $\mathcal{C}'$ to a configuration of $m+3$ lines with a single $(m+1)$-fold point and a single triple point. This latter configuration has a unique symplectic isotopy class by Proposition 5.1. Therefore by Corollary 3.19, the configurations $\mathcal{C}'$ has a unique symplectic isotopy class containing complex curve representatives, and thus $\mathcal{C}$ does as well. $\square$

LEMMA 5.13. Let $\mathcal{C}'$ denote the configuration above, obtained from $\mathcal{C}$ by adding $m$ lines tangent to $\mathcal{C}$ at $P$ as well as one line through $P$ and $Q$. Then the configuration $\mathcal{C}'$ is birationally equivalent to a configuration of $m+3$ lines where $m+1$ intersect at a single point and the other two intersect the $m+1$st at a triple point. (See Figure 9.)

*Proof.* Blowing up at the singular point of $\mathcal{C}$ yields a smooth rational curve, the proper transform of $\mathcal{C}$, with $m$ points of tangency with the exceptional divisor $E_0$, of order $d_1, \dots, d_m$ respectively. Blow up $d_i$ times at the $i$th tangency, so that the proper transform of $E_0$ is disjoint from that of $\mathcal{C}$. Next, blow up $d-1$ times at the tangency of $\mathcal{C}$ with $L$, so that their proper transforms intersect transversely.

We obtain the configuration of rational curves represented in Figure 10.

The proper transform of $\mathcal{C}$ has self-intersection $+1 = d^2 - (d-1)^2 - 2(d-1)$, and is a smooth sphere. We identify it with $\mathbb{CP}^1$ using Theorem 3.6 and use this to determine the homology classes of the other curves in the configuration.

By Lemma 3.7, and their pairwise disjointness, the vertical $(-1)$-curves intersecting $\mathcal{C}$ must represent classes $h - e_0 - e_1^i$ for $i = 1, \dots, m$ with the final vertical $(-1)$-curve which also intersects $L$ in the class $h - e_0 - e_1^0$. The chains of $(-2)$-curves emanating from these $(-1)$-curves
<figure><img src="image_18.png" /><figcaption>FIGURE 9. The two birationally equivalent configurations from Lemma 5.13. On the left, the configuration $\mathcal{C}'$ coming from the configuration $\mathcal{C}$ of Proposition 5.12. On the right, the configuration of $m+3$ lines with one ($m+1$)-fold point and one triple point.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 10. The configuration of curves in the proof of Lemma 5.13. The long hooked curves are $(-1)$-spheres which will be blown down to obtain the line arrangement on the right of Figure 9.</figcaption></figure>

are fully determined by Lemma 3.12 to be $e_1^i - e_2^i, \dots, e_{d_i-1}^i - e_{d_i}^i$. The last chain of $(-2)$-curves is similarly $e_1^0 - e_2^0, \dots, e_{d-2}^0 - e_{d-1}^0$. The intersection relations imply $[E_0] = e_0 - e_1^0 - \dots - e_{d-1}^0$ and $[L] = h - e_1^1 - \dots - e_{d_1}^1 - \dots - e_1^m - \dots - e_{d_m}^m$. The proper transforms of the tangent lines represent the classes $e_{d_i}^i$ for $1 \le i \le m$, and the proper transform of the line through the two singular points represents the class $e_{d-1}^0$. Therefore, we can blow down these exceptional curves, and consequently blow down the entire chain of $(-2)$-curves as the end curve becomes an exceptional divisor.

The resulting configuration consists of $m+3$ symplectic lines coming from the proper transforms of $C, L$, and the $m+1$ vertical $(-1)$-curves. The first $m$ vertical $(-1)$-curves will have a common intersection point which is the image of the blow-down of $e_0$. Here $C, L$, and the last vertical $(-1)$-curve have a common triple intersection point before blowing down, so this is preserved in the proper transform. $\square$

This completes the proof of Theorem 1.5.
<figure><img src="image_18.png" /><figcaption>FIGURE 9. The two birationally equivalent configurations from Lemma 5.13. On the left, the configuration $\mathcal{C}'$ coming from the configuration $\mathcal{C}$ of Proposition 5.12. On the right, the configuration of $m+3$ lines with one ($m+1$)-fold point and one triple point.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 10. The configuration of curves in the proof of Lemma 5.13. The long hooked curves are $(-1)$-spheres which will be blown down to obtain the line arrangement on the right of Figure 9.</figcaption></figure>

are fully determined by Lemma 3.12 to be $e_1^i - e_2^i, \dots, e_{d_i-1}^i - e_{d_i}^i$. The last chain of $(-2)$-curves is similarly $e_1^0 - e_2^0, \dots, e_{d-2}^0 - e_{d-1}^0$. The intersection relations imply $[E_0] = e_0 - e_1^0 - \dots - e_{d-1}^0$ and $[L] = h - e_1^1 - \dots - e_{d_1}^1 - \dots - e_1^m - \dots - e_{d_m}^m$. The proper transforms of the tangent lines represent the classes $e_{d_i}^i$ for $1 \le i \le m$, and the proper transform of the line through the two singular points represents the class $e_{d-1}^0$. Therefore, we can blow down these exceptional curves, and consequently blow down the entire chain of $(-2)$-curves as the end curve becomes an exceptional divisor.

The resulting configuration consists of $m+3$ symplectic lines coming from the proper transforms of $C, L$, and the $m+1$ vertical $(-1)$-curves. The first $m$ vertical $(-1)$-curves will have a common intersection point which is the image of the blow-down of $e_0$. Here $C, L$, and the last vertical $(-1)$-curve have a common triple intersection point before blowing down, so this is preserved in the proper transform. $\square$

This completes the proof of Theorem 1.5.
<figure><img src="image_20.png" /><figcaption>FIGURE 11. The configuration $\mathcal{H}$ built from a conic with three tangent lines and three additional lines intersecting in triple points as shown.</figcaption></figure>

*Proof of Theorem 1.5.* Line arrangements of degrees up to six are proven to have a unique and non-empty isotopy class in Corollary 5.3. Configurations of one conic and up to three lines are dealt with in Corollary 5.4, whereas configurations of two conics, or two conics and a line are taken care of by Corollary 5.11, and Proposition 5.12 settles the case of a rational degree-$d$ curve with a unique singularity of multiplicity $[d-1]$ and a line with an order-$d$ tangency. $\square$

5.1.3 *Two more configurations.* Here we show two additional configurations have a unique symplectic isotopy class. The proofs require some new ideas along with the pseudoholomorphic techniques and birational transformations we have been relying on. For the first, we utilize a fixed point argument. For the second, we define certain branched covering maps from a singular curve to $\mathbb{CP}^1$ utilizing pseudoholomorphic tangent lines and pencils. We will need the uniqueness of these configurations to prove there are unique symplectic isotopy classes of certain cuspidal quintics in Propositions 7.11 and 7.12.

The first configuration, that we call $\mathcal{H}$, is composed of a conic inscribed in a triangle of tangent lines meeting in vertices $A, B$, and $C$, and three lines $\ell_1, \ell_2$, and $\ell_3$ through $A, B$, and $C$, respectively, such that their pairwise intersections are on the conic. See Figure 11.

**PROPOSITION 5.14.** *There is a unique equisingular symplectic isotopy class of the configuration $\mathcal{H}$.*

*Proof.* Consider the configuration $\mathcal{H}_0$ of a single symplectic conic with three distinct tangent lines. This configuration has a unique equisingular symplectic isotopy class by Corollary 5.4. Moreover, for each symplectic realization of $\mathcal{H}_0$ in $\mathbb{CP}^2$, the space of almost complex structures $J$ on $\mathbb{CP}^2$ for which the realization is $J$-holomorphic is non-empty and contractible by Lemma 3.4. Therefore, the space of pairs $(H_0, J_0)$ where $H_0$ is a $J_0$-holomorphic realization of $\mathcal{H}_0$ is a path-connected space.

For a fixed pair $(H_0, J_0)$, we prove that we can add $J_0$-holomorphic lines to $H_0$ to get a realization of $\mathcal{H}$ in exactly two ways, and the map $p: \{H \in \mathcal{H} \text{ symplectic}\} \to \{(H_0, J_0) \in \mathcal{H}_0 \times \mathcal{J}^\omega(H_0)\}$ is a double covering map. Then, to show that it is the connected cover rather than the trivial covering, we show that for a particular pair $(H_0^{\text{std}}, i)$, the two extensions to complex realizations of $\mathcal{H}$ are symplectically isotopic. Therefore, any two symplectic realizations $H_1, H_2$ of $\mathcal{H}$ can be equisingularly symplectically isotoped each to one of the two extensions of $(H_0^{\text{std}}, i)$ and these two extensions are equisingularly symplectically isotopic, so $H_1$ and $H_2$ are as well.
<figure><img src="image_20.png" /><figcaption>FIGURE 11. The configuration $\mathcal{H}$ built from a conic with three tangent lines and three additional lines intersecting in triple points as shown.</figcaption></figure>

*Proof of Theorem 1.5.* Line arrangements of degrees up to six are proven to have a unique and non-empty isotopy class in Corollary 5.3. Configurations of one conic and up to three lines are dealt with in Corollary 5.4, whereas configurations of two conics, or two conics and a line are taken care of by Corollary 5.11, and Proposition 5.12 settles the case of a rational degree-$d$ curve with a unique singularity of multiplicity $[d-1]$ and a line with an order-$d$ tangency. $\square$

5.1.3 *Two more configurations.* Here we show two additional configurations have a unique symplectic isotopy class. The proofs require some new ideas along with the pseudoholomorphic techniques and birational transformations we have been relying on. For the first, we utilize a fixed point argument. For the second, we define certain branched covering maps from a singular curve to $\mathbb{CP}^1$ utilizing pseudoholomorphic tangent lines and pencils. We will need the uniqueness of these configurations to prove there are unique symplectic isotopy classes of certain cuspidal quintics in Propositions 7.11 and 7.12.

The first configuration, that we call $\mathcal{H}$, is composed of a conic inscribed in a triangle of tangent lines meeting in vertices $A, B$, and $C$, and three lines $\ell_1, \ell_2$, and $\ell_3$ through $A, B$, and $C$, respectively, such that their pairwise intersections are on the conic. See Figure 11.

**PROPOSITION 5.14.** *There is a unique equisingular symplectic isotopy class of the configuration $\mathcal{H}$.*

*Proof.* Consider the configuration $\mathcal{H}_0$ of a single symplectic conic with three distinct tangent lines. This configuration has a unique equisingular symplectic isotopy class by Corollary 5.4. Moreover, for each symplectic realization of $\mathcal{H}_0$ in $\mathbb{CP}^2$, the space of almost complex structures $J$ on $\mathbb{CP}^2$ for which the realization is $J$-holomorphic is non-empty and contractible by Lemma 3.4. Therefore, the space of pairs $(H_0, J_0)$ where $H_0$ is a $J_0$-holomorphic realization of $\mathcal{H}_0$ is a path-connected space.

For a fixed pair $(H_0, J_0)$, we prove that we can add $J_0$-holomorphic lines to $H_0$ to get a realization of $\mathcal{H}$ in exactly two ways, and the map $p: \{H \in \mathcal{H} \text{ symplectic}\} \to \{(H_0, J_0) \in \mathcal{H}_0 \times \mathcal{J}^\omega(H_0)\}$ is a double covering map. Then, to show that it is the connected cover rather than the trivial covering, we show that for a particular pair $(H_0^{\text{std}}, i)$, the two extensions to complex realizations of $\mathcal{H}$ are symplectically isotopic. Therefore, any two symplectic realizations $H_1, H_2$ of $\mathcal{H}$ can be equisingularly symplectically isotoped each to one of the two extensions of $(H_0^{\text{std}}, i)$ and these two extensions are equisingularly symplectically isotopic, so $H_1$ and $H_2$ are as well.
For a given $B = (H_0, J_0)$, let $Q$ denote the conic in $H_0$, and let $p_1, p_2, p_3$ denote the three points of intersection of the three tangent lines. Consider the three pencils of lines $\pi_i : \mathbb{CP}^2 \setminus \{p_i\} \to \ell_i$ through the points $p_i, i = 1, 2, 3$, where $\ell_i$ are $J_0$-holomorphic lines not containing $p_i$. Restricting these pencils to the conic $Q$, gives a branched double covering map $\pi_i|_Q : Q \to \ell_i$ with exactly two branch points. The map is degree two because a generic line intersects the conic $Q$ in two points. There are two branch points, each corresponding to a point $q \in Q$ where the $J_0$-holomorphic line through $q$ and $p_i$ is tangent to $Q$, by the Riemann-Hurwitz formula: $2 = 2(2) - \sum_{q \in Q} (e_{\pi_i}(q) - 1)$.

Let $f_i^B : Q \to Q$ denote the unique involution such that $\pi_i \circ f_i^B = \pi_i|_Q$. Note that $f_i^B$ is independent of the choice of the auxiliary line $\ell_i$. Informally, $f_i^B(q)$ is the other intersection of the line $\ell_{p_i,q}$ through $p_i$ and $q$ with $Q$; if $\ell_{p_i,q}$ is tangent to $Q$, then $f_i^B(q) = q$.

Consider now the composition $f_3^B \circ f_2^B \circ f_1^B : Q \to Q$: this is a complex automorphism of $Q$, which therefore has two fixed points $a_1$ and $a_2$. Note that if $q$ is one of the tangent points between $Q$ and a line in $H_0$, then $q$ is not a fixed point of $f_3^B \circ f_2^B \circ f_1^B$ because each tangency point is fixed by exactly two of the $f_i^B$ and is sent to a different point by the third $f_i^B$.

Add to $H_0$ $J_0$-holomorphic lines $L_1$ through $p_1$ and $a$, $L_2$ through $p_2$ and $f_1^B(a)$, and $L_3$ through $p_3$ and $f_2^B(f_1^B(a))$. Note that $L_3$ intersects $L_1$ at a point on $Q$ if and only if $a \in \{a_1, a_2\}$. Therefore each realization of $\mathcal{H}$ which is $J_0$-holomorphic and whose conic and tangent lines agree with $H_0$, is given by $H \cup L_1 \cup L_2 \cup L_3$ where the $L_i$ are defined using $a \in \{a_1, a_2\}$. This tells us that the preimage of a point in the map $p : \{H \in \mathcal{H} \text{ symplectic}\} \to \{(H_0, J_0) \in \mathcal{H}_0 \times \mathcal{J}^\omega(H_0)\}$ is two realizations of $\mathcal{H}$. Moreover, the constructions of $a_1, a_2$ and $L_1, L_2, L_3$ depend continuously on $J_0$, the conic $Q$, and the points $p_1, p_2, p_3$, which depend continuously on $H_0 \in \mathcal{H}_0$, so $p$ is a covering map.

Finally, we consider the relation between two explicit complex realizations of $\mathcal{H}$ with the same underlying $H_0$. Let $H_0$ be the following realization of $\mathcal{H}_0$. Start with three lines in $\mathbb{R}^2$ intersecting to form an equilateral triangle $ABC$ with side length $\overline{AB} = 1$ in the real Euclidian plane, and its inscribed circle $Q$, and then we complexify the configuration. For $\theta \in (0, \pi/3)$, consider the half-line $\ell_A^\theta$ starting from $A$, interior to the angle $BAC$, and such that the angle between $\ell_A$ and $AB$ is $\theta$. Analogously, define $\ell_B^\theta$ and $\ell_C^\theta$. Let $X = \ell_B^\theta \cap \ell_C^\theta$, $Y = \ell_C^\theta \cap \ell_A^\theta$, and $Z = \ell_A^\theta \cap \ell_B^\theta$ (here we drop the dependence on $\theta$ for convenience). We claim that there is a unique $\theta_0 \in (0, \pi/6)$ such that $X, Y$, and $Z$ all lie on $Q$ and that for $\pi/3 - \theta_0$ the corresponding points all lie on $Q$, too. The area of the triangle $XYZ$ is given by

$$ 
\begin{aligned}
A_{XYZ} &= A_{ABC} - A_{ABZ} - A_{BCX} - A_{CAY} = A_{ABC} - 3A_{ABZ} \\
&= \frac{\sqrt{3}}{4} - 3 \cdot \frac{1}{2} \overline{AX} \cdot \overline{AB} \cdot \sin \theta = \frac{\sqrt{3}}{4} - 3 \cdot \frac{1}{2} \cdot \frac{\overline{AB} \cdot \sin \left( \frac{\pi}{3} - \theta \right)}{\sin \frac{2\pi}{3}} \cdot \overline{AB} \cdot \sin \theta \\
&= \frac{\sqrt{3}}{4} - \sqrt{3} \cdot \sin \left( \frac{\pi}{3} - \theta \right) \sin \theta = \frac{\sqrt{3}}{4} \left( 1 - 2 \cos \left( \frac{\pi}{3} - 2\theta \right) + 2 \cos \left( \frac{\pi}{3} \right) \right) \\
&= \frac{\sqrt{3}}{2} \left( 1 - \cos \left( \frac{\pi}{3} - 2\theta \right) \right)
\end{aligned}
$$

and the latter is a continuous, decreasing function from $[0, \pi/6]$ to $[0, +\infty)$, taking values $\sqrt{3}/4$ at $\theta = 0$ and $0$ at $\theta = \pi/6$. This means that there is a unique value of $0 < \theta_0 < \pi/6$ such that the area of $XYZ$ is the area of the inscribed triangle in $Q$. By rotational symmetry, at $\theta_0$ the triangle $XYZ$ is in fact inscribed in $Q$, i.e. the configuration of $H = H_0 \cup \ell_A^{\theta_0} \cup \ell_B^{\theta_0} \cup \ell_C^{\theta_0}$ is a realization of $\mathcal{H}$. We observe that a reflection across one of the axes of the triangle preserves
For a given $B = (H_0, J_0)$, let $Q$ denote the conic in $H_0$, and let $p_1, p_2, p_3$ denote the three points of intersection of the three tangent lines. Consider the three pencils of lines $\pi_i : \mathbb{CP}^2 \setminus \{p_i\} \to \ell_i$ through the points $p_i, i = 1, 2, 3$, where $\ell_i$ are $J_0$-holomorphic lines not containing $p_i$. Restricting these pencils to the conic $Q$, gives a branched double covering map $\pi_i|_Q : Q \to \ell_i$ with exactly two branch points. The map is degree two because a generic line intersects the conic $Q$ in two points. There are two branch points, each corresponding to a point $q \in Q$ where the $J_0$-holomorphic line through $q$ and $p_i$ is tangent to $Q$, by the Riemann-Hurwitz formula: $2 = 2(2) - \sum_{q \in Q} (e_{\pi_i}(q) - 1)$.

Let $f_i^B : Q \to Q$ denote the unique involution such that $\pi_i \circ f_i^B = \pi_i|_Q$. Note that $f_i^B$ is independent of the choice of the auxiliary line $\ell_i$. Informally, $f_i^B(q)$ is the other intersection of the line $\ell_{p_i,q}$ through $p_i$ and $q$ with $Q$; if $\ell_{p_i,q}$ is tangent to $Q$, then $f_i^B(q) = q$.

Consider now the composition $f_3^B \circ f_2^B \circ f_1^B : Q \to Q$: this is a complex automorphism of $Q$, which therefore has two fixed points $a_1$ and $a_2$. Note that if $q$ is one of the tangent points between $Q$ and a line in $H_0$, then $q$ is not a fixed point of $f_3^B \circ f_2^B \circ f_1^B$ because each tangency point is fixed by exactly two of the $f_i^B$ and is sent to a different point by the third $f_i^B$.

Add to $H_0$ $J_0$-holomorphic lines $L_1$ through $p_1$ and $a$, $L_2$ through $p_2$ and $f_1^B(a)$, and $L_3$ through $p_3$ and $f_2^B(f_1^B(a))$. Note that $L_3$ intersects $L_1$ at a point on $Q$ if and only if $a \in \{a_1, a_2\}$. Therefore each realization of $\mathcal{H}$ which is $J_0$-holomorphic and whose conic and tangent lines agree with $H_0$, is given by $H \cup L_1 \cup L_2 \cup L_3$ where the $L_i$ are defined using $a \in \{a_1, a_2\}$. This tells us that the preimage of a point in the map $p : \{H \in \mathcal{H} \text{ symplectic}\} \to \{(H_0, J_0) \in \mathcal{H}_0 \times \mathcal{J}^\omega(H_0)\}$ is two realizations of $\mathcal{H}$. Moreover, the constructions of $a_1, a_2$ and $L_1, L_2, L_3$ depend continuously on $J_0$, the conic $Q$, and the points $p_1, p_2, p_3$, which depend continuously on $H_0 \in \mathcal{H}_0$, so $p$ is a covering map.

Finally, we consider the relation between two explicit complex realizations of $\mathcal{H}$ with the same underlying $H_0$. Let $H_0$ be the following realization of $\mathcal{H}_0$. Start with three lines in $\mathbb{R}^2$ intersecting to form an equilateral triangle $ABC$ with side length $\overline{AB} = 1$ in the real Euclidian plane, and its inscribed circle $Q$, and then we complexify the configuration. For $\theta \in (0, \pi/3)$, consider the half-line $\ell_A^\theta$ starting from $A$, interior to the angle $BAC$, and such that the angle between $\ell_A$ and $AB$ is $\theta$. Analogously, define $\ell_B^\theta$ and $\ell_C^\theta$. Let $X = \ell_B^\theta \cap \ell_C^\theta$, $Y = \ell_C^\theta \cap \ell_A^\theta$, and $Z = \ell_A^\theta \cap \ell_B^\theta$ (here we drop the dependence on $\theta$ for convenience). We claim that there is a unique $\theta_0 \in (0, \pi/6)$ such that $X, Y$, and $Z$ all lie on $Q$ and that for $\pi/3 - \theta_0$ the corresponding points all lie on $Q$, too. The area of the triangle $XYZ$ is given by

$$ 
\begin{align*}
A_{XYZ} &= A_{ABC} - A_{ABZ} - A_{BCX} - A_{CAY} = A_{ABC} - 3A_{ABZ} \\
&= \frac{\sqrt{3}}{4} - 3 \cdot \frac{1}{2} \overline{AX} \cdot \overline{AB} \cdot \sin \theta = \frac{\sqrt{3}}{4} - 3 \cdot \frac{1}{2} \cdot \frac{\overline{AB} \cdot \sin \left(\frac{\pi}{3} - \theta\right)}{\sin \frac{2\pi}{3}} \cdot \overline{AB} \cdot \sin \theta \\
&= \frac{\sqrt{3}}{4} - \sqrt{3} \cdot \sin \left(\frac{\pi}{3} - \theta\right) \sin \theta = \frac{\sqrt{3}}{4} \left(1 - 2 \cos \left(\frac{\pi}{3} - 2\theta\right) + 2 \cos \left(\frac{\pi}{3}\right)\right) \\
&= \frac{\sqrt{3}}{2} \left(1 - \cos \left(\frac{\pi}{3} - 2\theta\right)\right)
\end{align*}
$$

and the latter is a continuous, decreasing function from $[0, \pi/6]$ to $[0, +\infty)$, taking values $\sqrt{3}/4$ at $\theta = 0$ and $0$ at $\theta = \pi/6$. This means that there is a unique value of $0 < \theta_0 < \pi/6$ such that the area of $XYZ$ is the area of the inscribed triangle in $Q$. By rotational symmetry, at $\theta_0$ the triangle $XYZ$ is in fact inscribed in $Q$, i.e. the configuration of $H = H_0 \cup \ell_A^{\theta_0} \cup \ell_B^{\theta_0} \cup \ell_C^{\theta_0}$ is a realization of $\mathcal{H}$. We observe that a reflection across one of the axes of the triangle preserves
<figure><img src="image_21.png" /><figcaption>FIGURE 12. The configuration $\mathcal{L}$ of two conics and three lines. There are two simple tangency points between the conics not visible in this real picture.</figcaption></figure>

the triangle and the inscribed circle. As it preserves incidences, it sends the configuration $H$ to a configuration $H'$ that realizes $\mathcal{H}$ (indeed, $H'$ corresponds to the solution $\theta_1 = \pi/3 - \theta_0$; note that $\theta_1 > \pi/6$). We can extend the reflection to a complex linear isometry in $PU(3)$. As $PU(3)$ is a path-connected subset of $\text{Symp}(\mathbb{CP}^2)$, there is a family of symplectomorphisms carrying $H$ to $H'$ and tracing out a symplectic isotopy between them. $\square$

The second configuration, that we call $\mathcal{L}$, is made up of two conics $Q_1, Q_2$, tangent at two points, together with three lines such that each line is tangent to $Q_1$ and the pairwise intersections of the lines are three distinct points on $Q_2$. See Figure 12.

**PROPOSITION 5.15.** *There is a unique non-empty equisingular symplectic isotopy class of the configuration $\mathcal{L}$.*

*Proof.* We define two auxiliary configurations, $\mathcal{L}'$ and $\mathcal{L}'_0$. The latter, $\mathcal{L}'_0$, consists of a tricuspidal quartic $\mathcal{V}$ and the triangle of lines passing transversally through its singularities. The former, $\mathcal{L}'$, is obtained from $\mathcal{L}'_0$ by adding a bitangent to $\mathcal{V}$, i.e. a line that is tangent to $\mathcal{V}$ at two distinct points, each of which is a smooth point of $\mathcal{V}$.

First, observe that $\mathcal{L}$ is birationally equivalent to the configuration $\mathcal{L}'$. This is seen by blowing up once at each vertex of the triangle formed by the three lines, and then blowing down the proper transforms of the three lines (which are $(-1)$-spheres).

Next, we see that $\mathcal{L}'_0$ has a unique symplectic realization, up to isotopy. Using the inverse birational equivalence just described, we see that $\mathcal{L}'_0$ is birationally equivalent to the configuration $\mathcal{H}_0$ of a conic inscribed in a triangle of lines, which has a unique realization, up to isotopy, by Corollary 5.4.

What we really want to show is that $\mathcal{L}'$ has a unique symplectic isotopy realization, which follows from showing that there is a unique way, up to symplectic isotopy, to add the bitangent line to $\mathcal{L}'_0$ to obtain $\mathcal{L}'$. This is shown in the following proposition. Then because of the birational equivalence above, $\mathcal{L}$ itself will have a unique non-empty equisingular isotopy class. $\square$

**PROPOSITION 5.16.** *In any symplectic realization $V \cup L_1 \cup L_2 \cup L_3$ of $\mathcal{L}'_0$, the tricuspidal quartic $V$ has a unique bitangent up to isotopy, and this bitangent necessarily intersects the lines $L_i$ in generic transverse double points. Therefore, $\mathcal{L}'$ has a unique symplectic realization.*

*Proof.* Fix a tricuspidal quartic $V \subset \mathbb{CP}^2$, and call $V_0$ the set of smooth points of $V$. Let $n : \mathbb{CP}^1 \to V$ be the normalization map. Fix an almost complex structure $J$ compatible with $\omega$ such that $V$ is $J$-holomorphic. Note that the set of such choices is contractible by Lemma 3.4. We prove that
<figure><img src="image_21.png" /><figcaption>FIGURE 12. The configuration $\mathcal{L}$ of two conics and three lines. There are two simple tangency points between the conics not visible in this real picture.</figcaption></figure>

the triangle and the inscribed circle. As it preserves incidences, it sends the configuration $H$ to a configuration $H'$ that realizes $\mathcal{H}$ (indeed, $H'$ corresponds to the solution $\theta_1 = \pi/3 - \theta_0$; note that $\theta_1 > \pi/6$). We can extend the reflection to a complex linear isometry in $PU(3)$. As $PU(3)$ is a path-connected subset of $\text{Symp}(\mathbb{CP}^2)$, there is a family of symplectomorphisms carrying $H$ to $H'$ and tracing out a symplectic isotopy between them. $\square$

The second configuration, that we call $\mathcal{L}$, is made up of two conics $Q_1, Q_2$, tangent at two points, together with three lines such that each line is tangent to $Q_1$ and the pairwise intersections of the lines are three distinct points on $Q_2$. See Figure 12.

**PROPOSITION 5.15.** *There is a unique non-empty equisingular symplectic isotopy class of the configuration $\mathcal{L}$.*

*Proof.* We define two auxiliary configurations, $\mathcal{L}'$ and $\mathcal{L}'_0$. The latter, $\mathcal{L}'_0$, consists of a tricuspidal quartic $\mathcal{V}$ and the triangle of lines passing transversally through its singularities. The former, $\mathcal{L}'$, is obtained from $\mathcal{L}'_0$ by adding a bitangent to $\mathcal{V}$, i.e. a line that is tangent to $\mathcal{V}$ at two distinct points, each of which is a smooth point of $\mathcal{V}$.

First, observe that $\mathcal{L}$ is birationally equivalent to the configuration $\mathcal{L}'$. This is seen by blowing up once at each vertex of the triangle formed by the three lines, and then blowing down the proper transforms of the three lines (which are $(-1)$-spheres).

Next, we see that $\mathcal{L}'_0$ has a unique symplectic realization, up to isotopy. Using the inverse birational equivalence just described, we see that $\mathcal{L}'_0$ is birationally equivalent to the configuration $\mathcal{H}_0$ of a conic inscribed in a triangle of lines, which has a unique realization, up to isotopy, by Corollary 5.4.

What we really want to show is that $\mathcal{L}'$ has a unique symplectic isotopy realization, which follows from showing that there is a unique way, up to symplectic isotopy, to add the bitangent line to $\mathcal{L}'_0$ to obtain $\mathcal{L}'$. This is shown in the following proposition. Then because of the birational equivalence above, $\mathcal{L}$ itself will have a unique non-empty equisingular isotopy class. $\square$

**PROPOSITION 5.16.** *In any symplectic realization $V \cup L_1 \cup L_2 \cup L_3$ of $\mathcal{L}'_0$, the tricuspidal quartic $V$ has a unique bitangent up to isotopy, and this bitangent necessarily intersects the lines $L_i$ in generic transverse double points. Therefore, $\mathcal{L}'$ has a unique symplectic realization.*

*Proof.* Fix a tricuspidal quartic $V \subset \mathbb{CP}^2$, and call $V_0$ the set of smooth points of $V$. Let $n : \mathbb{CP}^1 \to V$ be the normalization map. Fix an almost complex structure $J$ compatible with $\omega$ such that $V$ is $J$-holomorphic. Note that the set of such choices is contractible by Lemma 3.4. We prove that
there is a unique $J$-holomorphic bitangent line to $V$. As this will hold for every $J$ compatible with $V$, we will have a unique $J$-holomorphic realization of $\mathcal{L}'$ whose tricuspidal quartic agrees with $V$ for each $J \in \mathcal{J}^\omega(V)$. Varying $J$ through this contractible (and thus path-connected) space yields equisingular isotopies fixing $V$ and isotoping the lines (the triangle of lines $L_1, L_2, L_3$ through the cusps and the bitangent line) in the configuration of $\mathcal{L}'$ by defining the lines $L_i$ and $T$ as the unique $J_t$-holomorphic lines with the specified intersection properties. Therefore, any two symplectic extensions of $V$ to a realization of $\mathcal{L}'$ will be related by an equisingular symplectic isotopy.

From now on, we fix a $J$ which makes $VJ$-holomorphic, and all of our choices of curves will be $J$-holomorphic. Fix a generic $J$-holomorphic line $\ell$ to be the target of a pencil-like map.

First, we associate to each point $p \in V_0$, a $J$-holomorphic line $\ell_p$ through $p$ with the property that $\ell_p$ is tangent to $V$ at another point $q \neq p$, $q \in V$. We prove that such $\ell_p$ exists and is unique in Lemma 5.17. By tangency at $q$, we mean that either $q \in V_0$ and $\ell_p$ is tangent to $V$ at $q$, or that $q$ is a cusp of $V$, and $\ell_p$ is the tangent to the cusp. We also note that, in fact, the line $\ell_p$ might also be tangent to $V$ at $p$, in which case it is a bitangent to $V$.

Next, define a map $\phi_0: V_0 \to \ell$ by $p \mapsto \ell_p \cap \ell$. Extend this map to $V$ by letting, for each cusp $p$ of $V$, $\ell_p$ be the tangent to $V$ at the cusp. We can show that this extension gives a continuous map $\phi_1: V \to \ell$ as follows. For any point $q$ in a neighborhood of a cusp $p$, let $T_q$ be the unique $J$-holomorphic tangent line to $V$ at $q$. Then the family of submanifolds $T_q$ varies continuously with $q$. Therefore, the intersection points with multiplicities $T_q \cap V$ also vary continuously with $q$. When $q \neq p$, $T_q \cap V = \{q, p_q, r_q\}$ where $q$ has multiplicity two and $p_q$ and $r_q$ have multiplicity one. When $q = p$, $T_p \cap V = \{p, r\}$ where $p$ has multiplicity three and $r$ has multiplicity one. Therefore, we must have $r_q \to r$ as $q \to p$ and $p_q \to p$ as $q \to p$. Now, let $N$ be a small closed neighborhood of $p$ in $V$, and consider the map $f: N \to V$ defined by $f(q) = p_q$ (we choose $N$ sufficiently small to be able to distinguish $p_q$ (the points which converge to $p$ as $q \to p$) from $r_q$ (the points which converge to $r$ as $q \to p$)). Then $f$ is a continuous map and $f(p) = p$. In addition, $f$ is injective because if $f(q) = f(q') = p_q$, then the $J$-holomorphic pencil based at $p_q$ would include a tangent line to $V$ at $q$ and a tangent line to $V$ at $q'$, but such a pencil can only include a single tangent line to $V$ by Lemma 5.17. As $N$ is compact, and $f$ is continuous and injective, $f$ is a homeomorphism onto its image. Therefore, $f(N) \subset V$ is homeomorphic to a disk centered at $p$. In particular, there is a neighborhood $U$ of $p$ such that for all $p'_t \in U$, $\ell_{p'_t}$ is tangent to $V$ inside $N$ so $\ell_{p'_t} = T_{q_t}$ for some $q_t \in N$. Therefore, if $p'_t \to p$, $\ell_{p'_t} = T_{q_t} \to T_p$.

Let $\phi = \phi_1 \circ n: \mathbb{CP}^1 \to \ell$. We show that $\phi: \mathbb{CP}^1 \to \ell$ is a branched covering of degree six and analyze all the ways in which ramification points can arise.

Given a point $s \in \ell \setminus V$, $\phi^{-1}(s)$ consists of points $p$ such that the unique $J$-holomorphic line through $s$ and $p$ is tangent to $V$ at a point different from $p$ (i.e. $\ell_p$ passes through $s$). Therefore, we need to understand the $J$-holomorphic lines in the pencil based at $s$ which have tangencies to $V$. As $s \notin V$, the Riemann–Hurwitz formula gives us that for this pencil, the sum of the ramification numbers corresponding to precomposing this pencil with the normalization is $\sum (e(x_j) - 1) = 6$. Each of the three cusps will be a ramification point $n(x_j)$ for some $j$. We have $e(x_j) = 2$ if the line in the pencil through the cusp $n(x_j)$ is transverse to $V$ at $n(x_j)$, and $e(x_j)$ is at least three if the line in the pencil through the cusp $n(x_j)$ is tangent to $V$ at $n(x_j)$. Therefore, we could have in the pencil through $s$:

(a) three distinct tangent lines to $V$, each intersecting $V$ transversally at two other points (or if the tangency is at a cusp, intersecting $V$ only at one other point transversally, but counting the cusp as the other point);
there is a unique $J$-holomorphic bitangent line to $V$. As this will hold for every $J$ compatible with $V$, we will have a unique $J$-holomorphic realization of $\mathcal{L}'$ whose tricuspidal quartic agrees with $V$ for each $J \in \mathcal{J}^\omega(V)$. Varying $J$ through this contractible (and thus path-connected) space yields equisingular isotopies fixing $V$ and isotoping the lines (the triangle of lines $L_1, L_2, L_3$ through the cusps and the bitangent line) in the configuration of $\mathcal{L}'$ by defining the lines $L_i$ and $T$ as the unique $J_t$-holomorphic lines with the specified intersection properties. Therefore, any two symplectic extensions of $V$ to a realization of $\mathcal{L}'$ will be related by an equisingular symplectic isotopy.

From now on, we fix a $J$ which makes $VJ$-holomorphic, and all of our choices of curves will be $J$-holomorphic. Fix a generic $J$-holomorphic line $\ell$ to be the target of a pencil-like map.

First, we associate to each point $p \in V_0$, a $J$-holomorphic line $\ell_p$ through $p$ with the property that $\ell_p$ is tangent to $V$ at another point $q \neq p$, $q \in V$. We prove that such $\ell_p$ exists and is unique in Lemma 5.17. By tangency at $q$, we mean that either $q \in V_0$ and $\ell_p$ is tangent to $V$ at $q$, or that $q$ is a cusp of $V$, and $\ell_p$ is the tangent to the cusp. We also note that, in fact, the line $\ell_p$ might also be tangent to $V$ at $p$, in which case it is a bitangent to $V$.

Next, define a map $\phi_0: V_0 \to \ell$ by $p \mapsto \ell_p \cap \ell$. Extend this map to $V$ by letting, for each cusp $p$ of $V$, $\ell_p$ be the tangent to $V$ at the cusp. We can show that this extension gives a continuous map $\phi_1: V \to \ell$ as follows. For any point $q$ in a neighborhood of a cusp $p$, let $T_q$ be the unique $J$-holomorphic tangent line to $V$ at $q$. Then the family of submanifolds $T_q$ varies continuously with $q$. Therefore, the intersection points with multiplicities $T_q \cap V$ also vary continuously with $q$. When $q \neq p$, $T_q \cap V = \{q, p_q, r_q\}$ where $q$ has multiplicity two and $p_q$ and $r_q$ have multiplicity one. When $q = p$, $T_p \cap V = \{p, r\}$ where $p$ has multiplicity three and $r$ has multiplicity one. Therefore, we must have $r_q \to r$ as $q \to p$ and $p_q \to p$ as $q \to p$. Now, let $N$ be a small closed neighborhood of $p$ in $V$, and consider the map $f: N \to V$ defined by $f(q) = p_q$ (we choose $N$ sufficiently small to be able to distinguish $p_q$ (the points which converge to $p$ as $q \to p$) from $r_q$ (the points which converge to $r$ as $q \to p$)). Then $f$ is a continuous map and $f(p) = p$. In addition, $f$ is injective because if $f(q) = f(q') = p_q$, then the $J$-holomorphic pencil based at $p_q$ would include a tangent line to $V$ at $q$ and a tangent line to $V$ at $q'$, but such a pencil can only include a single tangent line to $V$ by Lemma 5.17. As $N$ is compact, and $f$ is continuous and injective, $f$ is a homeomorphism onto its image. Therefore, $f(N) \subset V$ is homeomorphic to a disk centered at $p$. In particular, there is a neighborhood $U$ of $p$ such that for all $p'_t \in U$, $\ell_{p'_t}$ is tangent to $V$ inside $N$ so $\ell_{p'_t} = T_{q_t}$ for some $q_t \in N$. Therefore, if $p'_t \to p$, $\ell_{p'_t} = T_{q_t} \to T_p$.

Let $\phi = \phi_1 \circ n: \mathbb{CP}^1 \to \ell$. We show that $\phi: \mathbb{CP}^1 \to \ell$ is a branched covering of degree six and analyze all the ways in which ramification points can arise.

Given a point $s \in \ell \setminus V$, $\phi^{-1}(s)$ consists of points $p$ such that the unique $J$-holomorphic line through $s$ and $p$ is tangent to $V$ at a point different from $p$ (i.e. $\ell_p$ passes through $s$). Therefore, we need to understand the $J$-holomorphic lines in the pencil based at $s$ which have tangencies to $V$. As $s \notin V$, the Riemann–Hurwitz formula gives us that for this pencil, the sum of the ramification numbers corresponding to precomposing this pencil with the normalization is $\sum (e(x_j) - 1) = 6$. Each of the three cusps will be a ramification point $n(x_j)$ for some $j$. We have $e(x_j) = 2$ if the line in the pencil through the cusp $n(x_j)$ is transverse to $V$ at $n(x_j)$, and $e(x_j)$ is at least three if the line in the pencil through the cusp $n(x_j)$ is tangent to $V$ at $n(x_j)$. Therefore, we could have in the pencil through $s$:

(a) three distinct tangent lines to $V$, each intersecting $V$ transversally at two other points (or if the tangency is at a cusp, intersecting $V$ only at one other point transversally, but counting the cusp as the other point);
(b) one bitangent line to V and one tangent line with two transverse intersections to V; or

(c) at least one line in the pencil has a higher order tangency to V.

(Note that there cannot be a line with three distinct tangencies to V because V only has degree four and three tangencies would contribute six to the intersection.)

In the first case (the generic case), we have six points in $\phi^{-1}(s)$ coming from the six transverse intersections of the tangent lines with V. (In the case that one of the tangent lines is tangent at the cusp, the two points are the cusp point and the transverse intersection.) As transversality is an open condition, any such $s$ has an open neighborhood $U$ such that the pencil through $s' \in U$ falls into the first case. After possibly shrinking $U$, we can show $\phi^{-1}(U)$ is homeomorphic to $U \times \{1, 2, 3, 4, 5, 6\}$. To show this, let $q \in V$ be a point such that the line from $q$ to $s$ is tangent to $V$ at $q$. For $q_t$ converging to $q$, the tangent line to $V$ at $q_t$ intersects $\ell$ at points $s_t$ covering to $s$, and intersects $V$ at points $p_t^1$ and $p_t^2$ converging to $p^1, p^2 \in \phi^{-1}(s)$. Because the number of tangent lines to $V$ in any given pencil is finite, we can ensure that if $q_t$ is sufficiently close to $q$, then $p_t^i \neq p_{t'}^i$ for $i=1, 2$, and $s_t \neq s_{t'}$ for $t \neq t'$. Therefore, the maps $q_t \mapsto s_t$ and $q_t \mapsto p_t^i$ are continuous and injective maps, so they locally have continuous inverses. As $\phi(p_t) = s_t$, we see that $\phi$ is a covering map of degree six at generic points.

In the second case, the bitangent line intersects $V$ at the two tangent points, $p_0, p_1$. Let $\tau_0$ denote the $J$-line in the pencil through $s$ which is bitangent to $V$ and let $\tau_1$ denote the $J$-line in the pencil through $s$ which is simply tangent to $V$. Therefore, by the definition of $\phi$, $\phi^{-1}(s)$ consists of the two transverse intersections of $\tau_1$ with $V$ and the two tangential intersections $\tau_0 \cap V$. Thus, there are only four preimage points instead of six so the contribution to the Euler characteristic reduction is $\sum_{x \in \phi^{-1}(s)} (e_\phi(x) - 1) = 2$.

The third case is ruled out as a possibility in Lemmas 5.18 and 5.19.

Next, consider a point $s \in \ell \cap V$ (there are four of them). It follows from Lemma 5.17 that there is a unique line $\ell_s$ which passes through $s$ and is tangent to $V$ at a different point $q \neq s$. Its two transverse intersections with $V$ ($s$ and one other point $p$) will be regular points in $\phi^{-1}(s)$ as in the generic case. The only other way that a line tangent to $V$ could pass through $s$ is if the tangency occurs at $s$. Let $T_s$ be the tangent $J$-line to $V$ at $s$. Then the only other points in $\phi^{-1}(s)$ are the two points where $T_s$ intersects $V$ transversally. (By the genericity of $\ell$, we ensure that $T_s$ intersects $V$ at two points transversally.) Therefore, $\phi^{-1}(s)$ consists of four points instead of six, so at each intersection $s \in \ell \cap V$, $\sum_{s \in \phi^{-1}(s)} (e_\phi(x) - 1) = 2$.

Finally, we apply the Riemann-Hurwitz formula to $\phi$:

$$2 = \chi(\mathbb{CP}^1) = 6\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\phi(x) - 1) = 12 - 8 - \sum_{x \in \mathbb{CP}^1 \setminus n^{-1}(\ell)} (e_\phi(x) - 1).$$

Therefore, $\sum_{x \in \mathbb{CP}^1 \setminus n^{-1}(\ell)} (e_\phi(x) - 1) = 2$. As the only contribution to ramification indices for points in $\phi^{-1}(\ell \setminus V)$ is a bitangent, and each bitangent contributes two to this sum, we conclude that there is a unique $J$-holomorphic bitangent line to $V$ as claimed. $\square$

**LEMMA 5.17.** For every point $p \in V_0$ there is a unique line $\ell_p$ through $p$ that has a tangency to $V$ at another point $q \neq p$.

*Proof.* Consider the $J$-linear pencil based at $p: \pi_0: \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$. Restrict $\pi_0$ to $V \setminus \{p\}$, and extend it to $\pi_1: V \to \mathbb{CP}^1$ by defining $\pi_1(p) = T_p$, where $T_p$ is the tangent to $V$ at $p$. The map $\pi_1$ is continuous: the tangent $T_p$ is the only curve that has local multiplicity of intersection two with $C$ at $p$, and the lines $\ell_{pq}$ through $p$ and $q$ converge to such a curve as $q$ limits to $p$. Finally, define $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ by $\pi = \pi_1 \circ n$, where $n: \mathbb{CP}^1 \to V$ is the normalization. By positivity of intersections, $\pi$ is a branched cover $\mathbb{CP}^1 \to \mathbb{CP}^1$ of degree $4-1=3$. Moreover, if $n(x)$ is a cusp
(b) one bitangent line to V and one tangent line with two transverse intersections to V; or

(c) at least one line in the pencil has a higher order tangency to V.

(Note that there cannot be a line with three distinct tangencies to V because V only has degree four and three tangencies would contribute six to the intersection.)

In the first case (the generic case), we have six points in $\phi^{-1}(s)$ coming from the six transverse intersections of the tangent lines with V. (In the case that one of the tangent lines is tangent at the cusp, the two points are the cusp point and the transverse intersection.) As transversality is an open condition, any such $s$ has an open neighborhood $U$ such that the pencil through $s' \in U$ falls into the first case. After possibly shrinking $U$, we can show $\phi^{-1}(U)$ is homeomorphic to $U \times \{1, 2, 3, 4, 5, 6\}$. To show this, let $q \in V$ be a point such that the line from $q$ to $s$ is tangent to V at $q$. For $q_t$ converging to $q$, the tangent line to $V$ at $q_t$ intersects $\ell$ at points $s_t$ covering to $s$, and intersects $V$ at points $p_t^1$ and $p_t^2$ converging to $p^1, p^2 \in \phi^{-1}(s)$. Because the number of tangent lines to $V$ in any given pencil is finite, we can ensure that if $q_t$ is sufficiently close to $q$, then $p_t^i \neq p_{t'}^i$ for $i=1, 2$, and $s_t \neq s_{t'}$ for $t \neq t'$. Therefore, the maps $q_t \mapsto s_t$ and $q_t \mapsto p_t^i$ are continuous and injective maps, so they locally have continuous inverses. As $\phi(p_t) = s_t$, we see that $\phi$ is a covering map of degree six at generic points.

In the second case, the bitangent line intersects V at the two tangent points, $p_0, p_1$. Let $\tau_0$ denote the J-line in the pencil through $s$ which is bitangent to V and let $\tau_1$ denote the J-line in the pencil through $s$ which is simply tangent to V. Therefore, by the definition of $\phi$, $\phi^{-1}(s)$ consists of the two transverse intersections of $\tau_1$ with V and the two tangential intersections $\tau_0 \cap V$. Thus, there are only four preimage points instead of six so the contribution to the Euler characteristic reduction is $\sum_{x \in \phi^{-1}(s)} (e_\phi(x) - 1) = 2$.

The third case is ruled out as a possibility in Lemmas 5.18 and 5.19.

Next, consider a point $s \in \ell \cap V$ (there are four of them). It follows from Lemma 5.17 that there is a unique line $\ell_s$ which passes through $s$ and is tangent to $V$ at a different point $q \neq s$. Its two transverse intersections with $V$ ($s$ and one other point $p$) will be regular points in $\phi^{-1}(s)$ as in the generic case. The only other way that a line tangent to $V$ could pass through $s$ is if the tangency occurs at $s$. Let $T_s$ be the tangent J-line to $V$ at $s$. Then the only other points in $\phi^{-1}(s)$ are the two points where $T_s$ intersects $V$ transversally. (By the genericity of $\ell$, we ensure that $T_s$ intersects $V$ at two points transversally.) Therefore, $\phi^{-1}(s)$ consists of four points instead of six, so at each intersection $s \in \ell \cap V$, $\sum_{s \in \phi^{-1}(s)} (e_\phi(x) - 1) = 2$.

Finally, we apply the Riemann-Hurwitz formula to $\phi$:

$$2 = \chi(\mathbb{CP}^1) = 6\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\phi(x) - 1) = 12 - 8 - \sum_{x \in \mathbb{CP}^1 \setminus n^{-1}(\ell)} (e_\phi(x) - 1).$$

Therefore, $\sum_{x \in \mathbb{CP}^1 \setminus n^{-1}(\ell)} (e_\phi(x) - 1) = 2$. As the only contribution to ramification indices for points in $\phi^{-1}(\ell \setminus V)$ is a bitangent, and each bitangent contributes two to this sum, we conclude that there is a unique J-holomorphic bitangent line to V as claimed. $\square$

**LEMMA 5.17.** For every point $p \in V_0$ there is a unique line $\ell_p$ through $p$ that has a tangency to $V$ at another point $q \neq p$.

*Proof.* Consider the J-linear pencil based at $p: \pi_0: \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$. Restrict $\pi_0$ to $V \setminus \{p\}$, and extend it to $\pi_1: V \to \mathbb{CP}^1$ by defining $\pi_1(p) = T_p$, where $T_p$ is the tangent to $V$ at $p$. The map $\pi_1$ is continuous: the tangent $T_p$ is the only curve that has local multiplicity of intersection two with $C$ at $p$, and the lines $\ell_{pq}$ through $p$ and $q$ converge to such a curve as $q$ limits to $p$. Finally, define $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ by $\pi = \pi_1 \circ n$, where $n: \mathbb{CP}^1 \to V$ is the normalization. By positivity of intersections, $\pi$ is a branched cover $\mathbb{CP}^1 \to \mathbb{CP}^1$ of degree $4-1=3$. Moreover, if $n(x)$ is a cusp
of V, then $x$ is a ramification point of $x$. The index of $x$ as a ramification point is two if $p$ is not the intersection of the tangent to $\pi(x)$ with $V$, and three otherwise.

In the former case, the Riemann–Hurwitz formula yields

$$2 = \chi(\mathbb{CP}^1) = 3\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\pi(x) - 1) = 6 - 3 - \sum_{n(x) \in V_0} (e_\pi(x) - 1),$$

which implies that there is exactly one point $x$ in $\mathbb{CP}^1$ such that $\pi$ has ramification of index two at $x$. This means exactly that $\ell_p$ is tangent to $V$ at $n(x)$.

In the latter case, Riemann–Hurwitz gives

$$2 = \chi(\mathbb{CP}^1) = 3\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\pi(x) - 1) = 6 - 2 - 1 - 1 - \sum_{n(x) \in V_0} (e_\pi(x) - 1),$$

which means that the only ramification points of $\pi$ are at the preimages of cusps of $V$. This means that the only tangent to $V$ through $p$ is the tangent to the cusp.

We exclude the possibility that $n^{-1}(p)$ itself is a ramification point of $\pi$ in Lemma 5.18. If this occurred, $T_p$ would be a simple inflection line of $V$, which is exactly what the lemma obstructs. □

**LEMMA 5.18.** *The tricuspidal quartic V has no simple inflection J-lines at non-singular points.*

*Proof.* Suppose $p \in V_0$ is a simple inflection point, and let $q$ be the unique other intersection of $T_p$ and $Q$. Observe that, by positivity of intersections, $q \in V_0$. Now define the branched covering map $\pi' : \mathbb{CP}^1 \to \mathbb{CP}^1$ as in Lemma 5.17, but use the point $q$ as the base of the pencil instead of $p$. This is a degree-three cover. Each of the three cusps has ramification index two, and the inflection point has ramification index at least three. However, using the Riemann–Hurwitz formula, we get a contradiction

$$2 = 3 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 6 - 1 - 1 - 1 - 2 = 1. \quad \square$$

**LEMMA 5.19.** *The tricuspidal quartic V has no multiplicity-four tangent J-lines.*

*Proof.* Suppose that there was such a J-line and call $p$ its (unique) intersection point with $V$. Here $p$ could either be a non-singular point, i.e. $p \in V_0$, or a cusp of $V$. In either case, we consider the pencil of J-lines through $p$ and project from $p$. Precomposing with the normalization of $V$, we get a branched cover $\pi' : \mathbb{CP}^1 \to \mathbb{CP}^1$.

In the first case, when $p$ is a non-singular point of $V$, $\pi'$ is a three-fold branched covering, with ramification points of index at least two at each of the three cusps and of index four at $p$ itself. This violates the Riemann–Hurwitz formula because we would have

$$2 = 3 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 6 - 3 - 1 - 1 - 1 = 0.$$

In the second case, when $p$ is a cusp, the considering the same pencil, $\pi'$ is a two-fold branched covering which has ramification points of index at least two at the two remaining cusps and exactly two at $p$, yielding a similar contradiction:

$$2 = 2 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 4 - 2 - 1 - 1 = 1. \quad \square$$

## 5.2 Obstructions

Next we show that certain singular configurations do not symplectically embed into $\mathbb{CP}^2$ using birational derivations, and the following result.
of V, then x is a ramification point of x. The index of x as a ramification point is two if p is not the intersection of the tangent to π(x) with V, and three otherwise.

In the former case, the Riemann–Hurwitz formula yields

$$2 = \chi(\mathbb{CP}^1) = 3\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\pi(x) - 1) = 6 - 3 - \sum_{n(x) \in V_0} (e_\pi(x) - 1),$$

which implies that there is exactly one point x in $\mathbb{CP}^1$ such that $\pi$ has ramification of index two at x. This means exactly that $\ell_p$ is tangent to V at $n(x)$.

In the latter case, Riemann–Hurwitz gives

$$2 = \chi(\mathbb{CP}^1) = 3\chi(\mathbb{CP}^1) - \sum_{x \in \mathbb{CP}^1} (e_\pi(x) - 1) = 6 - 2 - 1 - 1 - \sum_{n(x) \in V_0} (e_\pi(x) - 1),$$

which means that the only ramification points of $\pi$ are at the preimages of cusps of V. This means that the only tangent to V through p is the tangent to the cusp.

We exclude the possibility that $n^{-1}(p)$ itself is a ramification point of $\pi$ in Lemma 5.18. If this occurred, $T_p$ would be a simple inflection line of V, which is exactly what the lemma obstructs. □

**LEMMA 5.18.** *The tricuspidal quartic V has no simple inflection J-lines at non-singular points.*

*Proof.* Suppose $p \in V_0$ is a simple inflection point, and let $q$ be the unique other intersection of $T_p$ and $Q$. Observe that, by positivity of intersections, $q \in V_0$. Now define the branched covering map $\pi' : \mathbb{CP}^1 \to \mathbb{CP}^1$ as in Lemma 5.17, but use the point $q$ as the base of the pencil instead of $p$. This is a degree-three cover. Each of the three cusps has ramification index two, and the inflection point has ramification index at least three. However, using the Riemann–Hurwitz formula, we get a contradiction

$$2 = 3 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 6 - 1 - 1 - 1 - 2 = 1. \quad \square$$

**LEMMA 5.19.** *The tricuspidal quartic V has no multiplicity-four tangent J-lines.*

*Proof.* Suppose that there was such a J-line and call $p$ its (unique) intersection point with V. Here $p$ could either be a non-singular point, i.e. $p \in V_0$, or a cusp of V. In either case, we consider the pencil of J-lines through $p$ and project from $p$. Precomposing with the normalization of V, we get a branched cover $\pi' : \mathbb{CP}^1 \to \mathbb{CP}^1$.

In the first case, when $p$ is a non-singular point of V, $\pi'$ is a three-fold branched covering, with ramification points of index at least two at each of the three cusps and of index four at $p$ itself. This violates the Riemann–Hurwitz formula because we would have

$$2 = 3 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 6 - 3 - 1 - 1 - 1 = 0.$$

In the second case, when $p$ is a cusp, the considering the same pencil, $\pi'$ is a two-fold branched covering which has ramification points of index at least two at the two remaining cusps and exactly two at $p$, yielding a similar contradiction:

$$2 = 2 \cdot 2 - \sum_{x \in \mathbb{CP}^1} (e_{\pi'}(x) - 1) \le 4 - 2 - 1 - 1 = 1. \quad \square$$

## 5.2 Obstructions

Next we show that certain singular configurations do not symplectically embed into $\mathbb{CP}^2$ using birational derivations, and the following result.
<figure><img src="image_22.png" /><figcaption>FIGURE 13. The Fano configuration of seven lines intersecting in seven triple points.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 14. Configuration $\mathcal{G}$ consists of three lines intersecting at a common triple point, each tangent to a conic.</figcaption></figure>

THEOREM 5.20 [RS19, Theorem 1.1]. There is no symplectic embedding in $\mathbb{CP}^2$ of the Fano plane: seven lines intersecting positively at seven triple intersection points (see Figure 13).

The next configuration we consider is made up of three lines each tangent to a conic such that the three lines intersect each other at a triple point. We call this configuration $\mathcal{G}$ (Figure 14).

PROPOSITION 5.21. There is no symplectic embedding of $\mathcal{G}$ into $\mathbb{CP}^2$.

*Proof.* Let $Q$ be any smooth symplectic sphere in $\mathbb{CP}^2$ with $[Q] = 2h$. Let $T_1, T_2$, and $T_3$ be three symplectic tangent lines at points $p_1, p_2$, and $p_3$ on $Q$. Suppose $T_1, T_2$, and $T_3$ all intersect at a common triple point.

Follow Figure 15 with the rest of the proof. Blow up once at each of $p_1, p_2$, and $p_3$. Then the proper transform of $Q$ is a symplectic sphere of self-intersection +1 so by Theorem 3.6 it can be identified with $\mathbb{CP}^1$ with homology class $h$ via a symplectomorphism $\Psi$ of $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$. Under this identification, Lemma 3.7 and the intersections of the components determines the homology classes of the remaining curves as follows:

$$ \Psi_*[\overline{Q}] = h, \quad \Psi_*[\overline{T_i}] = h - e_i, \quad \Psi_*[E_i] = h - e_1 - e_2 - e_3 + e_i. $$

Next, by Lemma 3.5, we can blow down exceptional spheres in the homology classes $e_i$ such that any intersection with $\Psi(\overline{Q}), \Psi(\overline{T_i})$, or $\Psi(\overline{E_i})$ is positive. After blowing down three times, the proper transform in $\mathbb{CP}^2$ of each of the seven surfaces is in the homology class $h$ and there are the following triple intersections: $P_i = \Psi(\overline{Q}) \cap \Psi(E_i) \cap \Psi(\overline{T_i})$ for $i = 1, 2, 3$, $R_1 = \Psi(\overline{T_1}) \cap \Psi(E_2) \cap \Psi(E_3)$, $R_2 = \Psi(\overline{T_2}) \cap \Psi(E_1) \cap \Psi(E_3)$, $R_3 = \Psi(\overline{T_3}) \cap \Psi(E_1) \cap \Psi(E_2)$. If there were a seventh triple intersection between $\Psi(\overline{T_1}), \Psi(\overline{T_2})$, and $\Psi(\overline{T_3})$ this would give an embedding of the Fano configuration into $\mathbb{CP}^2$.

Thus, we have shown that the Fano plane is birationally derived from $\mathcal{G}$, so by the contrapositive of Proposition 3.17, there is no symplectic realization of $\mathcal{G}$ in $\mathbb{CP}^2$. (In fact, the
<figure><img src="image_22.png" /><figcaption>FIGURE 13. The Fano configuration of seven lines intersecting in seven triple points.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 14. Configuration $\mathcal{G}$ consists of three lines intersecting at a common triple point, each tangent to a conic.</figcaption></figure>

THEOREM 5.20 [RS19, Theorem 1.1]. There is no symplectic embedding in $\mathbb{CP}^2$ of the Fano plane: seven lines intersecting positively at seven triple intersection points (see Figure 13).

The next configuration we consider is made up of three lines each tangent to a conic such that the three lines intersect each other at a triple point. We call this configuration $\mathcal{G}$ (Figure 14).

PROPOSITION 5.21. There is no symplectic embedding of $\mathcal{G}$ into $\mathbb{CP}^2$.

*Proof.* Let $Q$ be any smooth symplectic sphere in $\mathbb{CP}^2$ with $[Q] = 2h$. Let $T_1, T_2$, and $T_3$ be three symplectic tangent lines at points $p_1, p_2$, and $p_3$ on $Q$. Suppose $T_1, T_2$, and $T_3$ all intersect at a common triple point.

Follow Figure 15 with the rest of the proof. Blow up once at each of $p_1, p_2$, and $p_3$. Then the proper transform of $Q$ is a symplectic sphere of self-intersection +1 so by Theorem 3.6 it can be identified with $\mathbb{CP}^1$ with homology class $h$ via a symplectomorphism $\Psi$ of $\mathbb{CP}^2\#3\overline{\mathbb{CP}}^2$. Under this identification, Lemma 3.7 and the intersections of the components determines the homology classes of the remaining curves as follows:

$$ \Psi_*[\overline{Q}] = h, \quad \Psi_*[\overline{T_i}] = h - e_i, \quad \Psi_*[E_i] = h - e_1 - e_2 - e_3 + e_i. $$

Next, by Lemma 3.5, we can blow down exceptional spheres in the homology classes $e_i$ such that any intersection with $\Psi(\overline{Q}), \Psi(\overline{T_i})$, or $\Psi(\overline{E_i})$ is positive. After blowing down three times, the proper transform in $\mathbb{CP}^2$ of each of the seven surfaces is in the homology class $h$ and there are the following triple intersections: $P_i = \Psi(\overline{Q}) \cap \Psi(E_i) \cap \Psi(\overline{T_i})$ for $i = 1, 2, 3$, $R_1 = \Psi(\overline{T_1}) \cap \Psi(E_2) \cap \Psi(E_3)$, $R_2 = \Psi(\overline{T_2}) \cap \Psi(E_1) \cap \Psi(E_3)$, $R_3 = \Psi(\overline{T_3}) \cap \Psi(E_1) \cap \Psi(E_2)$. If there were a seventh triple intersection between $\Psi(\overline{T_1}), \Psi(\overline{T_2})$, and $\Psi(\overline{T_3})$ this would give an embedding of the Fano configuration into $\mathbb{CP}^2$.

Thus, we have shown that the Fano plane is birationally derived from $\mathcal{G}$, so by the contrapositive of Proposition 3.17, there is no symplectic realization of $\mathcal{G}$ in $\mathbb{CP}^2$. (In fact, the
<figure><img src="image_24.png" /><figcaption>FIGURE 15. Birational derivation from $\mathcal{G}$ to the Fano configuration.</figcaption></figure>

same argument using the full strength of Theorem 3.6 shows that there is no symplectic realization of a configuration $\mathcal{G}$ with the same self-intersection numbers into any closed symplectic 4-manifold.) $\square$

Note that the exceptional spheres representing $e_1, e_2, e_3$ are not contained in the total transform of the embedding of $\mathcal{G}$, so the Fano plane is birationally derived from $\mathcal{G}$, but they are not birationally equivalent.

Here $\mathcal{G}$ is dual to the configuration $\mathcal{G}^*$, comprised of three conics in a pencil (intersecting at four triple points) and a line tangent to all three of them. We can also obstruct $\mathcal{G}^*$.

**PROPOSITION 5.22.** *There is no symplectic embedding of the configuration $\mathcal{G}^*$ in $\mathbb{CP}^2$.*

*Proof.* We show that there is a birational derivation from $\mathcal{G}^*$ to a configuration containing $\mathcal{G}$. Given a symplectic realization of $\mathcal{G}^*$, blow up at three of the four basepoints of the pencil. The proper transforms of the conics are symplectic +1-spheres, intersecting at a single point (namely, the fourth basepoint). We can apply Theorem 3.6 to identify one of the three conics with $\mathbb{CP}^1$. Then Lemma 3.7 implies the other two conics are also symplectic lines, the image of the tangent line has class $2h - e_1 - e_2 - e_3$, and the three exceptional spheres represent $h - e_1 - e_2$, $h - e_2 - e_3$, and $h - e_1 - e_3$. Blowing down positively intersecting exceptional spheres in classes $e_1, e_2, e_3$ using Lemma 3.5, the total transform of the realization of $\mathcal{G}^*$ blows down to a conic simultaneously tangent to each of the three concurrent lines, together with a triangle of lines inscribed in the conic. As this configuration contains $\mathcal{G}$, the configuration $\mathcal{G}^*$ is obstructed by Proposition 3.17. $\square$

We also obstruct the following related configurations:

$\mathcal{G}_4$: comprising two conics with a single point of tangency of order four, and a line tangent to both;

$\mathcal{G}_{2,2}$: comprising two conics with two simple tangencies, and a line tangent to both.

**PROPOSITION 5.23.** *There is no symplectic embedding of any of the configurations $\mathcal{G}_4$ or $\mathcal{G}_{2,2}$ in $\mathbb{CP}^2$.*
<figure><img src="image_24.png" /><figcaption>FIGURE 15. Birational derivation from $\mathcal{G}$ to the Fano configuration.</figcaption></figure>

same argument using the full strength of Theorem 3.6 shows that there is no symplectic realization of a configuration $\mathcal{G}$ with the same self-intersection numbers into any closed symplectic 4-manifold.) $\square$

Note that the exceptional spheres representing $e_1, e_2, e_3$ are not contained in the total transform of the embedding of $\mathcal{G}$, so the Fano plane is birationally derived from $\mathcal{G}$, but they are not birationally equivalent.

Here $\mathcal{G}$ is dual to the configuration $\mathcal{G}^*$, comprised of three conics in a pencil (intersecting at four triple points) and a line tangent to all three of them. We can also obstruct $\mathcal{G}^*$.

**PROPOSITION 5.22.** There is no symplectic embedding of the configuration $\mathcal{G}^*$ in $\mathbb{CP}^2$.

*Proof.* We show that there is a birational derivation from $\mathcal{G}^*$ to a configuration containing $\mathcal{G}$. Given a symplectic realization of $\mathcal{G}^*$, blow up at three of the four basepoints of the pencil. The proper transforms of the conics are symplectic +1-spheres, intersecting at a single point (namely, the fourth basepoint). We can apply Theorem 3.6 to identify one of the three conics with $\mathbb{CP}^1$. Then Lemma 3.7 implies the other two conics are also symplectic lines, the image of the tangent line has class $2h - e_1 - e_2 - e_3$, and the three exceptional spheres represent $h - e_1 - e_2$, $h - e_2 - e_3$, and $h - e_1 - e_3$. Blowing down positively intersecting exceptional spheres in classes $e_1, e_2, e_3$ using Lemma 3.5, the total transform of the realization of $\mathcal{G}^*$ blows down to a conic simultaneously tangent to each of the three concurrent lines, together with a triangle of lines inscribed in the conic. As this configuration contains $\mathcal{G}$, the configuration $\mathcal{G}^*$ is obstructed by Proposition 3.17. $\square$

We also obstruct the following related configurations:

$\mathcal{G}_4$: comprising two conics with a single point of tangency of order four, and a line tangent to both;

$\mathcal{G}_{2,2}$: comprising two conics with two simple tangencies, and a line tangent to both.

**PROPOSITION 5.23.** There is no symplectic embedding of any of the configurations $\mathcal{G}_4$ or $\mathcal{G}_{2,2}$ in $\mathbb{CP}^2$.
<figure><img src="image_25.png" /><figcaption>FIGURE 16. Birational derivation of $\mathcal{G}$ from $\mathcal{G}_4$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE 17. Birational derivation of $\mathcal{G}$ from $\mathcal{G}_{2,2}$.</figcaption></figure>

*Proof*. Suppose that there existed an embedding of $\mathcal{G}_4$ in $\mathbb{CP}^2$. After blowing up three times at the tangency point of the two conics, the proper transforms of the two conics are two $+1$-spheres. See Figure 16. Applying Theorem 3.6, we obtain a symplectomorphism of $\mathbb{CP}^2$ blown up three times that sends the proper transform of one of the conics to $\mathbb{CP}^1$. Using Lemma 3.7 and intersection numbers, the proper transforms of all the conics are in the homology class $h$, the proper transforms of the three exceptional divisors (in the order in which we made the blow-ups) are mapped to spheres in homology classes $e_1 - e_2$, $e_2 - e_3$, and $h - e_1 - e_2$, and the proper transform of the line is sent to the homology class $2h - e_1 - e_2 - e_3$. Using Lemma 3.5 to blow down a positively intersecting sphere in the class $e_3$ followed by the $(-1)$-spheres in the configuration representing $e_2$ and $e_1$, we get a birational derivation from $\mathcal{G}_4$ to $\mathcal{G}$.

For $\mathcal{G}_{2,2}$ we blow up twice at one of the tangencies, and once at the other. See Figure 17. Identifying the proper transforms of the conics with $\mathbb{CP}^1$, the homology classes of the other curves in the total transform are uniquely determined by Lemma 3.7 and intersection numbers. The proper transform of the line is again identified with a sphere in the class $2h - e_1 - e_2 - e_3$, and exceptional sphere at the intersection where we blew up once represents $h - e_1 - e_2$. The two exceptional curves from the other blow-ups are identified with spheres representing the classes $h - e_2 - e_3$ and $e_2 - e_1$. Using Lemma 3.5 to blow-down exceptional spheres in classes $e_1$, $e_2$, $e_3$, the configuration blows down to a configuration containing $\mathcal{G}$ (with an additional line). Therefore, by Proposition 3.17, a symplectic embedding of $\mathcal{G}_{2,2}$ into $\mathbb{CP}^2$ is obstructed. $\square$

*Remark 5.24.* The existence of a symplectic embedding of $\mathcal{G}$ into $\mathbb{CP}^2$ can also be obstructed using a $J$-holomorphic linear pencil based at the triple point $p$, and finding a contradiction
<figure><img src="image_25.png" /><figcaption>FIGURE 16. Birational derivation of $\mathcal{G}$ from $\mathcal{G}_4$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE 17. Birational derivation of $\mathcal{G}$ from $\mathcal{G}_{2,2}$.</figcaption></figure>

*Proof*. Suppose that there existed an embedding of $\mathcal{G}_4$ in $\mathbb{CP}^2$. After blowing up three times at the tangency point of the two conics, the proper transforms of the two conics are two $+1$-spheres. See Figure 16. Applying Theorem 3.6, we obtain a symplectomorphism of $\mathbb{CP}^2$ blown up three times that sends the proper transform of one of the conics to $\mathbb{CP}^1$. Using Lemma 3.7 and intersection numbers, the proper transforms of all the conics are in the homology class $h$, the proper transforms of the three exceptional divisors (in the order in which we made the blow-ups) are mapped to spheres in homology classes $e_1 - e_2$, $e_2 - e_3$, and $h - e_1 - e_2$, and the proper transform of the line is sent to the homology class $2h - e_1 - e_2 - e_3$. Using Lemma 3.5 to blow down a positively intersecting sphere in the class $e_3$ followed by the $(-1)$-spheres in the configuration representing $e_2$ and $e_1$, we get a birational derivation from $\mathcal{G}_4$ to $\mathcal{G}$.

For $\mathcal{G}_{2,2}$ we blow up twice at one of the tangencies, and once at the other. See Figure 17. Identifying the proper transforms of the conics with $\mathbb{CP}^1$, the homology classes of the other curves in the total transform are uniquely determined by Lemma 3.7 and intersection numbers. The proper transform of the line is again identified with a sphere in the class $2h - e_1 - e_2 - e_3$, and exceptional sphere at the intersection where we blew up once represents $h - e_1 - e_2$. The two exceptional curves from the other blow-ups are identified with spheres representing the classes $h - e_2 - e_3$ and $e_2 - e_1$. Using Lemma 3.5 to blow-down exceptional spheres in classes $e_1, e_2, e_3$, the configuration blows down to a configuration containing $\mathcal{G}$ (with an additional line). Therefore, by Proposition 3.17, a symplectic embedding of $\mathcal{G}_{2,2}$ into $\mathbb{CP}^2$ is obstructed. $\square$

*Remark 5.24.* The existence of a symplectic embedding of $\mathcal{G}$ into $\mathbb{CP}^2$ can also be obstructed using a $J$-holomorphic linear pencil based at the triple point $p$, and finding a contradiction
using Riemann-Hurwitz. Indeed, consider the projection $\pi: Q \to \mathbb{CP}^1$ induced by the pencil. By positivity of intersections, this is a branched covering map. Each tangent to $Q$ through $p$ gives a branching point of $\pi$ of index 2, and Riemann-Hurwitz yields

$$2 \cdot 2 = 2\chi(Q) = \chi(\mathbb{CP}^1) + \sum_{q \in Q} e_{\pi}(q) \geq 5,$$

a contradiction.

A similar argument also works for $\mathcal{G}^*$, $\mathcal{G}_4$, and $\mathcal{G}_{2,2}$.

## 6. Unicuspidal curves with one Newton pair

The goal of this section is to prove Theorem 1.2, that every symplectic unicuspidal curve in $\mathbb{CP}^2$ whose singularity has one Puiseux pair is isotopic to a complex curve. In fact, for each of such curve we classify all of its symplectic embeddings into any closed symplectic manifold. Correspondingly, we classify all the strong symplectic fillings of the corresponding contact manifolds.

According to [Liu14, Theorem 2.3] (see also [BCG16, Remark 6.18]), if a rational cuspidal curve with a unique singularity of type $(p, q)$ satisfies the adjunction formula, that is $(p-1)(q-1) = (d-1)(d-2)$, and the semigroup distribution property (2.4), then $(p, q)$ belongs to the list of [FLMN07, Theorem 1.1]; namely, $(p, q)$ is one of:

(i) $(p, p+1)$, with $p \geq 2$, and the curve has degree $p+1$;

(ii) $(p, 4p-1)$, with $p \geq 2$, and the curve has degree $2p$

(iii) $(F_{j-2}, F_{j+2})$, with $j \geq 5$ and odd, and the curve has degree $F_j$;

(iv) $(F_j^2; F_{j+2}^2)$ with $j \geq 3$ and odd, and the curve has degree $F_j F_{j+2}$;

(v) $(3, 22)$, and the curve has degree 8;

(vi) $(6, 43)$ and the curve has degree 16.

Here, $F_j$ denotes the $j$th Fibonacci number; recall that the sequence $\{F_j\}$ is defined by the recursion $F_{j+1} = F_j + F_{j-1}$, starting from $F_0 = 0, F_1 = 1$.

**Remark 6.1.** To be more precise, in [BCG16] the result is only stated asymptotically, i.e. for ‘sufficiently large’ degrees. (One can explicitly compute what ‘sufficiently large’ and verify that the result holds for all degrees. This is not mentioned in [BCG16].) In [Liu14], Liu explains in the paragraphs above Theorem 1.1 and above Theorem 2.3 that his results are obtained in a purely combinatorial way from the semigroup distribution property, which is a consequence of the Heegaard Floer restriction together with the adjunction formula. A more general result than what is needed here was obtained in [AGLL20], where the requirement on the adjunction formula is dropped. Sifting through the lists of [AGLL20, Theorem 1.1], one verifies that the only ones that satisfy the adjunction formula are those mentioned above.

As symplectic curves satisfy the adjunction formula (2.2) and the semigroup distribution property (2.4), we can restrict to the six cases above. For each of them, we use a resolution to classify symplectic fillings of the associated contact 3-manifold, and correspondingly classify the symplectic embeddings of these cuspidal curves in closed symplectic manifolds up to symplectic isotopy.

In § 6.1, we describe the choice of resolutions that we use to classify the embeddings. Each such resolution will contain a smooth symplectic +1-sphere as the proper transform of the cuspidal curve. In § 6.2, we use McDuff’s theorem and the lemmas of § 3.3 to classify all homological embeddings of each of the resolutions. In this story, the two Fibonacci families will play a
using Riemann-Hurwitz. Indeed, consider the projection $\pi : Q \to \mathbb{CP}^1$ induced by the pencil. By positivity of intersections, this is a branched covering map. Each tangent to $Q$ through $p$ gives a branching point of $\pi$ of index 2, and Riemann-Hurwitz yields

$$2 \cdot 2 = 2\chi(Q) = \chi(\mathbb{CP}^1) + \sum_{q \in Q} e_{\pi}(q) \geq 5,$$

a contradiction.

A similar argument also works for $\mathcal{G}^*$, $\mathcal{G}_4$, and $\mathcal{G}_{2,2}$.

## 6. Unicuspidal curves with one Newton pair

The goal of this section is to prove Theorem 1.2, that every symplectic unicuspidal curve in $\mathbb{CP}^2$ whose singularity has one Puiseux pair is isotopic to a complex curve. In fact, for each of such curve we classify all of its symplectic embeddings into any closed symplectic manifold. Correspondingly, we classify all the strong symplectic fillings of the corresponding contact manifolds.

According to [Liu14, Theorem 2.3] (see also [BCG16, Remark 6.18]), if a rational cuspidal curve with a unique singularity of type $(p, q)$ satisfies the adjunction formula, that is $(p-1)(q-1) = (d-1)(d-2)$, and the semigroup distribution property (2.4), then $(p, q)$ belongs to the list of [FLMN07, Theorem 1.1]; namely, $(p, q)$ is one of:

(i) $(p, p+1)$, with $p \geq 2$, and the curve has degree $p+1$;

(ii) $(p, 4p-1)$, with $p \geq 2$, and the curve has degree $2p$

(iii) $(F_{j-2}, F_{j+2})$, with $j \geq 5$ and odd, and the curve has degree $F_j$;

(iv) $(F_j^2; F_{j+2}^2)$ with $j \geq 3$ and odd, and the curve has degree $F_j F_{j+2}$;

(v) $(3, 22)$, and the curve has degree 8;

(vi) $(6, 43)$ and the curve has degree 16.

Here, $F_j$ denotes the $j$th Fibonacci number; recall that the sequence $\{F_j\}$ is defined by the recursion $F_{j+1} = F_j + F_{j-1}$, starting from $F_0 = 0, F_1 = 1$.

**Remark 6.1.** To be more precise, in [BCG16] the result is only stated asymptotically, i.e. for ‘sufficiently large’ degrees. (One can explicitly compute what ‘sufficiently large’ and verify that the result holds for all degrees. This is not mentioned in [BCG16].) In [Liu14], Liu explains in the paragraphs above Theorem 1.1 and above Theorem 2.3 that his results are obtained in a purely combinatorial way from the semigroup distribution property, which is a consequence of the Heegaard Floer restriction together with the adjunction formula. A more general result than what is needed here was obtained in [AGLL20], where the requirement on the adjunction formula is dropped. Sifting through the lists of [AGLL20, Theorem 1.1], one verifies that the only ones that satisfy the adjunction formula are those mentioned above.

As symplectic curves satisfy the adjunction formula (2.2) and the semigroup distribution property (2.4), we can restrict to the six cases above. For each of them, we use a resolution to classify symplectic fillings of the associated contact 3-manifold, and correspondingly classify the symplectic embeddings of these cuspidal curves in closed symplectic manifolds up to symplectic isotopy.

In § 6.1, we describe the choice of resolutions that we use to classify the embeddings. Each such resolution will contain a smooth symplectic +1-sphere as the proper transform of the cuspidal curve. In § 6.2, we use McDuff’s theorem and the lemmas of § 3.3 to classify all homological embeddings of each of the resolutions. In this story, the two Fibonacci families will play a
<figure><img src="image_27.png" /><figcaption>FIGURE 18. The top and bottom legs correspond to the continued fraction expansions $(p+1)/1 = [p+1]^{-}$ and $p/(p-1) = [2^{[p-1]}]^{-}$, respectively. The chain on the right is artificially longer than needed to make a +1-sphere appear.</figcaption></figure>

different role; we treat them in § 6.3. In § 6.4, we look at geometric realization of these homological embeddings, and prove a strengthening of Theorem 1.2. Finally, in § 6.5, we talk about rational blow-down relations.

## 6.1 The resolutions

Here we describe our preferred resolutions for the singular curves in each of cases (i)–(vi), except the Fibonacci families (for which we have a different argument in § 6.3). In general, our preferred resolution will not be either the minimal or the normal crossing divisor resolution of the singularity; the goal is to find a smooth symplectic +1-sphere.

We start with the first family: a degree-(p + 1) curve with a unique singularity of type (p, p + 1). We first find the normal crossing resolution. In the notation of § 2.1, we have $q = p + 1$, $p^* = p$ (because $p^2 \equiv 1 \pmod{p+1}$) and $q^* = 1$. We are interested in the continued fraction expansions of $p/(p-1)$ and $(p+1)/(p+1-p)$, which are $[2^{[p-1]}]^{-}$ and $[p+1]^{-}$, respectively. As the curve has degree p + 1, its self-intersection is $(p+1)^2$ and, therefore, the self-intersection of the proper transform in the normal crossing divisor resolution has self-intersection $(p+1)^2 - p(p+1) = p+1$. From this normal crossing resolution, we blow up p additional times along this third leg$^2$ (i.e. at the intersection between the latest exceptional divisor and the proper transform of the curve). The net effect of this operation is that the central vertex is decorated by Euler class -2 and the third leg consists of a chain of $(p-1)$ (-2)-vertices, a (-1)-vertex, and a (+1)-vertex. We refer to this symplectic plumbing as $\mathcal{A}_p$; see Figure 18.

We now turn to the second case: a degree-2p curve with a singularity of type (p, 4p − 1). The normal crossing resolution of this unicuspidal curve has two legs with expansions of p/1 = [p]− and (4p − 1)/(4p − 5) = [2[p−2], 3, 2, 2]−, and the third leg has one vertex, which is initially decorated by (2p)$^2$ − p(4p − 1) = p. We blow up along this leg p − 1 additional times, so that the central vertex is decorated by −2, and the third leg becomes a chain of p − 2 vertices decorated by −2, one by −1, and one by +1. We refer to this symplectic plumbing as $\mathcal{B}_p$.

The normal crossing resolution for the singularity of type (3, 22) has two legs expanding $\frac{3}{2} = [2, 2]^{-}$ and $\frac{22}{7} = [4, 2^{[6]}]^{-}$, with the third leg decorated by $8^2 - 3 \cdot 22 = -2$. As we want a sphere of square +1, we blow back down to the minimal resolution, whose neighborhood will be called $\mathcal{E}_3$. See Figure 19. The triple edge indicates a multiplicity-three tangency between the (+1)-sphere and the (-1)-sphere.

The normal crossing resolution for (6, 43) has two legs corresponding to the continued fraction expansions $\frac{6}{5} = [2^5]^-$ and $\frac{43}{7} = [7, 2^{[6]}]^-$, and the third leg is labeled by $16^2 - 6 \cdot 43 = -2$. Again we must blow down three times to get a (+1)-sphere. We end up between the minimal smooth

<footnote>$^2$What we call ‘leg’ here is usually called a ‘wig’ in algebraic geometry. We stick to the terminology of low-dimensional topology here.</footnote>
<figure><img src="image_27.png" /><figcaption>FIGURE 18. The top and bottom legs correspond to the continued fraction expansions $(p+1)/1 = [p+1]^{-}$ and $p/(p-1) = [2^{[p-1]}]^{-}$, respectively. The chain on the right is artificially longer than needed to make a +1-sphere appear.</figcaption></figure>

different role; we treat them in § 6.3. In § 6.4, we look at geometric realization of these homological embeddings, and prove a strengthening of Theorem 1.2. Finally, in § 6.5, we talk about rational blow-down relations.

## 6.1 The resolutions

Here we describe our preferred resolutions for the singular curves in each of cases (i)–(vi), except the Fibonacci families (for which we have a different argument in § 6.3). In general, our preferred resolution will not be either the minimal or the normal crossing divisor resolution of the singularity; the goal is to find a smooth symplectic +1-sphere.

We start with the first family: a degree-(p + 1) curve with a unique singularity of type (p, p + 1). We first find the normal crossing resolution. In the notation of § 2.1, we have $q = p + 1$, $p^* = p$ (because $p^2 \equiv 1 \pmod{p+1}$) and $q^* = 1$. We are interested in the continued fraction expansions of $p/(p-1)$ and $(p+1)/(p+1-p)$, which are $[2^{[p-1]}]^{-}$ and $[p+1]^{-}$, respectively. As the curve has degree $p+1$, its self-intersection is $(p+1)^2$ and, therefore, the self-intersection of the proper transform in the normal crossing divisor resolution has self-intersection $(p+1)^2 - p(p+1) = p+1$. From this normal crossing resolution, we blow up $p$ additional times along this third leg$^2$ (i.e. at the intersection between the latest exceptional divisor and the proper transform of the curve). The net effect of this operation is that the central vertex is decorated by Euler class $-2$ and the third leg consists of a chain of $(p-1)$ $(-2)$-vertices, a $(-1)$-vertex, and a $(+1)$-vertex. We refer to this symplectic plumbing as $\mathcal{A}_p$; see Figure 18.

We now turn to the second case: a degree-2p curve with a singularity of type $(p, 4p-1)$. The normal crossing resolution of this unicuspidal curve has two legs with expansions of $p/1 = [p]^{-}$ and $(4p-1)/(4p-5) = [2^{[p-2]}], 3, 2, 2]^{-}$, and the third leg has one vertex, which is initially decorated by $(2p)^2 - p(4p-1) = p$. We blow up along this leg $p-1$ additional times, so that the central vertex is decorated by $-2$, and the third leg becomes a chain of $p-2$ vertices decorated by $-2$, one by $-1$, and one by $+1$. We refer to this symplectic plumbing as $\mathcal{B}_p$.

The normal crossing resolution for the singularity of type (3, 22) has two legs expanding $\frac{3}{2} = [2, 2]^{-}$ and $\frac{22}{7} = [4, 2^{[6]}]^{-}$, with the third leg decorated by $8^2 - 3 \cdot 22 = -2$. As we want a sphere of square $+1$, we blow back down to the minimal resolution, whose neighborhood will be called $\mathcal{E}_3$. See Figure 19. The triple edge indicates a multiplicity-three tangency between the $(+1)$-sphere and the $(-1)$-sphere.

The normal crossing resolution for (6, 43) has two legs corresponding to the continued fraction expansions $\frac{6}{5} = [2^5]^-$ and $\frac{43}{7} = [7, 2^6]^-$, and the third leg is labeled by $16^2 - 6 \cdot 43 = -2$. Again we must blow down three times to get a (+1)-sphere. We end up between the minimal smooth

<footnote>$^2$ What we call ‘leg’ here is usually called a ‘wig’ in algebraic geometry. We stick to the terminology of low-dimensional topology here.</footnote>
M. GOLLA AND L. STARKSTON

$$ \mathcal{E}_3 = \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-1} \equiv \bullet^{+1} $$

<caption>FIGURE 19. The resolution graph $\mathcal{E}_3$.</caption>

$$ \mathcal{E}_6 = \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-4} \equiv \bullet^{+1} $$

<caption>FIGURE 20. The resolution graph $\mathcal{E}_6$.</caption>

resolution and the minimal normal crossing resolution, with a configuration indicated by the graph of Figure 20. We call the neighborhood $\mathcal{E}_6$. Here the (+1)- and (-4)-spheres intersect tangentially with multiplicity three, and the (-1)-sphere intersects these two at the same point transversally.

## 6.2 The homological embeddings

In this subsection, we apply McDuff's theorem to each of the resolutions of the previous subsection. In order to shorten up the statements in this section, using Theorem 3.6 we implicitly identify the (+1)-sphere in each of the configurations with a line in a blow-up of $\mathbb{CP}^2$, and correspondingly its homology class will be identified with $h$. We work with a standard basis for $H_2(\mathbb{CP}^2 \# N\overline{\mathbb{CP}^2}) = \langle h, e_1, \dots, e_N \rangle$. In each of the following statements and proofs describing the possible homology classes represented by components of a resolution, the uniqueness will always be up to a reordering of the $e_i$.

We start with the configuration $\mathcal{A}_p$, corresponding to curves with a singularity of type $(p, p+1)$.

LEMMA 6.2. In the configuration $\mathcal{A}_p$ of Figure 18, the homology classes of the curves in the chain which excludes the $(-p-1)$-sphere are

$$ (h, h - e_0 - e_1, e_1 - e_2, \dots, e_{p-1} - e_p, e_p - e_{p+1}, e_{p+1} - e_{p+2}, \dots, e_{2p-1} - e_{2p}) $$

and the symplectic $(-p-1)$-sphere represents

$$ e_0 - e_1 - \dots - e_p. $$

*Proof.* The classes of the components of the chain, excluding the $(-p-1)$-sphere, are uniquely determined by Lemma 3.12. The remaining $(-p-1)$-sphere intersects once positively with the class $e_p - e_{p+1}$ and zero with the other classes. As the class $e_{p+1}$ appears with positive coefficient already, by Lemmas 3.8–3.10 the class of this last sphere is uniquely determined as stated. $\square$

The cap corresponding to the rational cuspidal curve with singularity of type $(p, 4p-1)$ has similar restrictions on its possible homology classes.

LEMMA 6.3. In the configuration $\mathcal{B}_p$ of Figure 21, the long chain starting with the +1-sphere and including the entire configuration except the $(-p)$-sphere has two possible homology configurations differing only in the last sphere of the chain when $p \ge 3$. The first possibility is

$$ (h, h - e_0 - e_1, e_1 - e_2, \dots, e_{p-2} - e_{p-1}, e_{p-1} - e_p, e_p - e_{p+1}, \dots, e_{2p-3} - e_{2p-2}, \\ e_{2p-2} - e_{2p-1} - e_{2p}, e_{2p} - e_{2p+1}, e_{2p+1} - e_{2p+2}). $$

For the other possibility, the last sphere can represent the class $e_{2p-1} - e_{2p}$.
$$ \mathcal{E}_3 = \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-1} \equiv \bullet^{+1} $$

<caption>FIGURE 19. The resolution graph $\mathcal{E}_3$.</caption>

$$ \mathcal{E}_6 = \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-2} \bullet^{-4} \equiv \bullet^{+1} $$

<caption>FIGURE 20. The resolution graph $\mathcal{E}_6$.</caption>

resolution and the minimal normal crossing resolution, with a configuration indicated by the graph of Figure 20. We call the neighborhood $\mathcal{E}_6$. Here the (+1)- and (-4)-spheres intersect tangentially with multiplicity three, and the (-1)-sphere intersects these two at the same point transversally.

## 6.2 The homological embeddings

In this subsection, we apply McDuff’s theorem to each of the resolutions of the previous subsection. In order to shorten up the statements in this section, using Theorem 3.6 we implicitly identify the (+1)-sphere in each of the configurations with a line in a blow-up of $\mathbb{CP}^2$, and correspondingly its homology class will be identified with $h$. We work with a standard basis for $H_2(\mathbb{CP}^2\#N\overline{\mathbb{CP}^2}) = \langle h, e_1, \dots, e_N \rangle$. In each of the following statements and proofs describing the possible homology classes represented by components of a resolution, the uniqueness will always be up to a reordering of the $e_i$.

We start with the configuration $\mathcal{A}_p$, corresponding to curves with a singularity of type $(p, p+1)$.

LEMMA 6.2. In the configuration $\mathcal{A}_p$ of Figure 18, the homology classes of the curves in the chain which excludes the $(-p-1)$-sphere are

$$ (h, h - e_0 - e_1, e_1 - e_2, \dots, e_{p-1} - e_p, e_p - e_{p+1}, e_{p+1} - e_{p+2}, \dots, e_{2p-1} - e_{2p}) $$

and the symplectic $(-p-1)$-sphere represents

$$ e_0 - e_1 - \dots - e_p. $$

*Proof.* The classes of the components of the chain, excluding the $(-p-1)$-sphere, are uniquely determined by Lemma 3.12. The remaining $(-p-1)$-sphere intersects once positively with the class $e_p - e_{p+1}$ and zero with the other classes. As the class $e_{p+1}$ appears with positive coefficient already, by Lemmas 3.8–3.10 the class of this last sphere is uniquely determined as stated. $\square$

The cap corresponding to the rational cuspidal curve with singularity of type $(p, 4p-1)$ has similar restrictions on its possible homology classes.

LEMMA 6.3. In the configuration $\mathcal{B}_p$ of Figure 21, the long chain starting with the +1-sphere and including the entire configuration except the $(-p)$-sphere has two possible homology configurations differing only in the last sphere of the chain when $p \ge 3$. The first possibility is

$$ (h, h - e_0 - e_1, e_1 - e_2, \dots, e_{p-2} - e_{p-1}, e_{p-1} - e_p, e_p - e_{p+1}, \dots, e_{2p-3} - e_{2p-2}, \\ e_{2p-2} - e_{2p-1} - e_{2p}, e_{2p} - e_{2p+1}, e_{2p+1} - e_{2p+2}). $$

For the other possibility, the last sphere can represent the class $e_{2p-1} - e_{2p}$.
<figure><img src="image_28.png" /><figcaption>FIGURE 21. The top and bottom legs correspond to the continued fraction expansions $p/1 = [p]^{-}$ and $(4p-1)/(4p-5) = [2^{[p-2]}, 3, 2, 2]$, respectively. The chain on the right is artificially longer than needed to make a +1-sphere appear.</figcaption></figure>

In both cases, there is a unique possibility for the class represented by the $(-p)$-sphere:

$$e_0 - e_1 - \cdots - e_{p-1}.$$

When $p = 2$, there are the above two options as well as the following additional case where the long chain starting at the +1-sphere represents

$$(h, h - e_0 - e_1, e_1 - e_2, e_0 - e_1 - e_3, e_3 - e_4, e_4 - e_5)$$

and the length-one arm is a $(-p)$-sphere in the class $e_2 - e_6$.

Note that the second option is a homological embedding into $\mathbb{CP}^2 \# (2p + 2) \overline{\mathbb{CP}}^2$ whereas the first option is a homological embedding into $\mathbb{CP}^2 \# (2p + 3) \overline{\mathbb{CP}}^2$. As the cap has $b_2^+ = 1$ and $b_2^- = 2p + 2$, the symplectic filling complementary to the first embedding will have $b_2 = 1$ and that complementary to the second embedding will be a rational homology ball filling. In particular, the filling complementary to the second homology embedding is the only one that could give the complement of the rational cuspidal curve in $\mathbb{CP}^2$.

*Proof.* The $\mathcal{B}_p$ configuration is identical to the $\mathcal{A}_{p-1}$ configuration with three additional vertices of weights $-3, -2, -2$ on the lower left leg. Therefore, all of the homology classes excluding these last three, are determined by Lemma 6.2. By Lemmas 3.8 and 3.10, the $(-3)$-sphere must represent $e_{2p-2} - e_{2p-1} - e_{2p}$ unless $p = 2$ (we will come back to this case). When $p \ge 3$, the subsequent $(-2)$-sphere must represent $e_{2p} - e_{2p+1}$, and the last $(-2)$-sphere can either represent $e_{2p+1} - e_{2p+2}$ or $e_{2p-1} - e_{2p}$ by Lemma 3.11.

When $p = 2$, the $(-3)$-sphere can also represent $e_0 - e_1 - e_3$, and the remaining classes are determined as in the statement. $\square$

LEMMA 6.4. There are three homological embeddings of the symplectic configuration $\mathcal{E}_3$ into blow-ups of $\mathbb{CP}^2$ given as follows, listed linearly starting from the +1-sphere:

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_1 - e_7, e_7 - e_8, e_8 - e_9, e_9 - e_{10}, e_{10} - e_{11}, e_{11} - e_{12}), \qquad (6.1)$$

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_1 - e_7, e_2 - e_1, e_3 - e_2, e_4 - e_3, e_5 - e_4, e_6 - e_5), \qquad (6.2)$$

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6). \qquad (6.3)$$

*Proof.* The class of the $(-1)$-sphere which intersects $\mathbb{CP}^1$ with multiplicity three is determined by Lemma 3.7. The remaining chain of $(-2)$-spheres has one of two forms determined by Lemma 3.11. If it has form (B), except for the first sphere in the chain, all of the exceptional
<figure><img src="image_28.png" /><figcaption>FIGURE 21. The top and bottom legs correspond to the continued fraction expansions $p/1 = [p]^{-}$ and $(4p-1)/(4p-5) = [2^{[p-2]}, 3, 2, 2]$, respectively. The chain on the right is artificially longer than needed to make a +1-sphere appear.</figcaption></figure>

In both cases, there is a unique possibility for the class represented by the $(-p)$-sphere:

$$e_0 - e_1 - \cdots - e_{p-1}.$$

When $p = 2$, there are the above two options as well as the following additional case where the long chain starting at the +1-sphere represents

$$(h, h - e_0 - e_1, e_1 - e_2, e_0 - e_1 - e_3, e_3 - e_4, e_4 - e_5)$$

and the length-one arm is a $(-p)$-sphere in the class $e_2 - e_6$.

Note that the second option is a homological embedding into $\mathbb{CP}^2 \# (2p+2)\overline{\mathbb{CP}}^2$ whereas the first option is a homological embedding into $\mathbb{CP}^2 \# (2p+3)\overline{\mathbb{CP}}^2$. As the cap has $b_2^+ = 1$ and $b_2^- = 2p+2$, the symplectic filling complementary to the first embedding will have $b_2 = 1$ and that complementary to the second embedding will be a rational homology ball filling. In particular, the filling complementary to the second homology embedding is the only one that could give the complement of the rational cuspidal curve in $\mathbb{CP}^2$.

*Proof.* The $\mathcal{B}_p$ configuration is identical to the $\mathcal{A}_{p-1}$ configuration with three additional vertices of weights $-3, -2, -2$ on the lower left leg. Therefore, all of the homology classes excluding these last three, are determined by Lemma 6.2. By Lemmas 3.8 and 3.10, the $(-3)$-sphere must represent $e_{2p-2} - e_{2p-1} - e_{2p}$ unless $p = 2$ (we will come back to this case). When $p \ge 3$, the subsequent $(-2)$-sphere must represent $e_{2p} - e_{2p+1}$, and the last $(-2)$-sphere can either represent $e_{2p+1} - e_{2p+2}$ or $e_{2p-1} - e_{2p}$ by Lemma 3.11.

When $p = 2$, the $(-3)$-sphere can also represent $e_0 - e_1 - e_3$, and the remaining classes are determined as in the statement. $\square$

LEMMA 6.4. There are three homological embeddings of the symplectic configuration $\mathcal{E}_3$ into blow-ups of $\mathbb{CP}^2$ given as follows, listed linearly starting from the +1-sphere:

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_1 - e_7, e_7 - e_8, e_8 - e_9, e_9 - e_{10}, e_{10} - e_{11}, e_{11} - e_{12}), \qquad (6.1)$$

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_1 - e_7, e_2 - e_1, e_3 - e_2, e_4 - e_3, e_5 - e_4, e_6 - e_5), \qquad (6.2)$$

$$(h, 3h - 2e_0 - e_1 - e_2 - e_3 - e_4 - e_5 - e_6, e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6). \qquad (6.3)$$

*Proof.* The class of the $(-1)$-sphere which intersects $\mathbb{CP}^1$ with multiplicity three is determined by Lemma 3.7. The remaining chain of $(-2)$-spheres has one of two forms determined by Lemma 3.11. If it has form (B), except for the first sphere in the chain, all of the exceptional