classes appearing in the chain must have coefficient $-1$ in the previous sphere (Lemma 3.12), which uniquely yields option (6.2). If the chain has the form in Lemma 3.11 for option (A), the exceptional class with positive coefficient for the first $(-2)$-sphere is either $e_0$ or $e_1$ (without loss of generality). To ensure the correct intersection numbers, this uniquely determines options (6.3) and (6.1), respectively. □

The possibilities for embeddings of $\mathcal{E}_6$ are determined in the same way for the lower chain. The intersection numbers then limit the possibilities for the indices of the exceptional classes on the upper chain as follows.

LEMMA 6.5. There are six homological embeddings of the symplectic configuration $\mathcal{E}_6$ into blow-ups of $\mathbb{CP}^2$.

As in Lemma 6.4, there are three possibilities for the lower chain:

(i) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_1 - e_{10}, e_{10} - e_{11}, e_{11} - e_{12}, e_{12} - e_{13}, e_{13} - e_{14}, e_{14} - e_{15})$;

(ii) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_1 - e_{10}, e_2 - e_1, e_3 - e_2, e_4 - e_3, e_5 - e_4, e_6 - e_5)$;

(iii) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6)$.

When the lower chain is option (i), the upper chain can be

$$ (h - e_8 - e_9, e_8 - e_7, e_7 - e_6). \qquad (6.4) $$

When the lower chain is option (i) or (ii), the upper chain can be

$$ (h - e_0 - e_{16}, e_{16} - e_{17}, e_{17} - e_{18}) \text{ or } (h - e_0 - e_{11}, e_{11} - e_{12}, e_{12} - e_{13}), \text{ respectively.} \qquad (6.5) $$

When the lower chain is any of the three options, the upper chain can be

$$ (h - e_8 - e_9, e_8 - e_7, e_9 - e_8). \qquad (6.6) $$

## 6.3 The Fibonacci families

We start with the third family, $(F_{j-2}, F_{j+2})$ with $j$ odd. We recall that Fibonacci numbers satisfy the identity $F_j^2 = F_{j-2}F_{j+2} - 1$, so that by Moser [Mos71] the boundary $Y_j$ of a regular neighborhood of a curve in this family, with the orientation induced by the cuspidal contact structure $\xi_j$, is

$$ Y_j = -S_{F_j^2}^3(T(F_{j-2}, F_{j+2})) = -L(F_j^2, -F_{j-2}^2) = L(F_j^2, F_{j-2}^2). $$

Note that the identity $F_{j-2}^2 = F_{j-4}F_j - 1$ implies that $F_{j-2}^2 \equiv -1 \pmod{F_j}$, and that $\gcd(F_{j-2}, F_j) = F_{\gcd(j-2, j)} = F_1 = 1$, because $j$ is odd. In particular, $Y_j = L(m^2, mk - 1)$ for some $m, k$ with $\gcd(m, k) = 1$.

As there is a symplectic realization of a rational cuspidal curve in $\mathbb{CP}^2$ in this family, coming from algebraic geometry [FLMN07], $\xi_j$ has a rational homology ball symplectic filling. By Proposition A.1, $\xi_j$ is the canonical contact structure on $Y$ (perhaps up to conjugation), and by Proposition A.2, this filling is unique up to symplectic deformation.

We now turn to the fourth family, $(F_j^2, F_{j+2}^2)$, with $j$ odd. Call $(Y'_j, \xi'_j)$ the cuspidal contact manifold; by Moser [Mos71],

$$ Y'_j = -S_{F_j^2 F_{j+2}^2}^3 (T(F_j^2, F_{j+2}^2)) = -(L(F_j^2, -F_{j-2}^2) \# L(F_{j+2}^2, -F_j^2)) = Y_j \# Y_{j+2}. $$

As previously, by [FLMN07], $\xi'_j$ has a rational homology ball (strong) symplectic filling $W_0$; moreover, $(Y', \xi'_j)$ is the contact connected sum $(Y_j, \eta_j)\#(Y_{j+2}, \eta_{j+2})$. Note that $\xi'_j$ is planar, because all contact structures on lens spaces are; we recall that, by [Wen10], all strong symplectic fillings of planar contact structures can be deformed to Stein fillings, so that $W_0$ is
classes appearing in the chain must have coefficient $-1$ in the previous sphere (Lemma 3.12), which uniquely yields option (6.2). If the chain has the form in Lemma 3.11 for option (A), the exceptional class with positive coefficient for the first $(-2)$-sphere is either $e_0$ or $e_1$ (without loss of generality). To ensure the correct intersection numbers, this uniquely determines options (6.3) and (6.1), respectively. $\square$

The possibilities for embeddings of $\mathcal{E}_6$ are determined in the same way for the lower chain. The intersection numbers then limit the possibilities for the indices of the exceptional classes on the upper chain as follows.

LEMMA 6.5. There are six homological embeddings of the symplectic configuration $\mathcal{E}_6$ into blow-ups of $\mathbb{CP}^2$.

As in Lemma 6.4, there are three possibilities for the lower chain:

(i) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_1 - e_{10}, e_{10} - e_{11}, e_{11} - e_{12}, e_{12} - e_{13}, e_{13} - e_{14}, e_{14} - e_{15})$;

(ii) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_1 - e_{10}, e_2 - e_1, e_3 - e_2, e_4 - e_3, e_5 - e_4, e_6 - e_5)$;

(iii) $(h, 3h - 2e_0 - e_1 - \dots - e_9, e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6)$.

When the lower chain is option (i), the upper chain can be

$$ (h - e_8 - e_9, e_8 - e_7, e_7 - e_6). \qquad (6.4) $$

When the lower chain is option (i) or (ii), the upper chain can be

$$ (h - e_0 - e_{16}, e_{16} - e_{17}, e_{17} - e_{18}) \text{ or } (h - e_0 - e_{11}, e_{11} - e_{12}, e_{12} - e_{13}), \text{ respectively.} \qquad (6.5) $$

When the lower chain is any of the three options, the upper chain can be

$$ (h - e_8 - e_9, e_8 - e_7, e_9 - e_8). \qquad (6.6) $$

## 6.3 The Fibonacci families

We start with the third family, $(F_{j-2}, F_{j+2})$ with $j$ odd. We recall that Fibonacci numbers satisfy the identity $F_j^2 = F_{j-2}F_{j+2} - 1$, so that by Moser [Mos71] the boundary $Y_j$ of a regular neighborhood of a curve in this family, with the orientation induced by the cuspidal contact structure $\xi_j$, is

$$ Y_j = -S_{F_j^2}^3(T(F_{j-2}, F_{j+2})) = -L(F_j^2, -F_{j-2}^2) = L(F_j^2, F_{j-2}^2). $$

Note that the identity $F_{j-2}^2 = F_{j-4}F_j - 1$ implies that $F_{j-2}^2 \equiv -1 \pmod{F_j}$, and that $\gcd(F_{j-2}, F_j) = F_{\gcd(j-2, j)} = F_1 = 1$, because $j$ is odd. In particular, $Y_j = L(m^2, mk - 1)$ for some $m, k$ with $\gcd(m, k) = 1$.

As there is a symplectic realization of a rational cuspidal curve in $\mathbb{CP}^2$ in this family, coming from algebraic geometry [FLMN07], $\xi_j$ has a rational homology ball symplectic filling. By Proposition A.1, $\xi_j$ is the canonical contact structure on $Y$ (perhaps up to conjugation), and by Proposition A.2, this filling is unique up to symplectic deformation.

We now turn to the fourth family, $(F_j^2, F_{j+2}^2)$, with $j$ odd. Call $(Y'_j, \xi'_j)$ the cuspidal contact manifold; by Moser [Mos71],

$$ Y'_j = -S_{F_j^2 F_{j+2}^2}^3 (T(F_j^2, F_{j+2}^2)) = -(L(F_j^2, -F_{j-2}^2) \# L(F_{j+2}^2, -F_j^2)) = Y_j \# Y_{j+2}. $$

As previously, by [FLMN07], $\xi'_j$ has a rational homology ball (strong) symplectic filling $W_0$; moreover, $(Y', \xi'_j)$ is the contact connected sum $(Y_j, \eta_j)\#(Y_{j+2}, \eta_{j+2})$. Note that $\xi'_j$ is planar, because all contact structures on lens spaces are; we recall that, by [Wen10], all strong symplectic fillings of planar contact structures can be deformed to Stein fillings, so that $W_0$ is
in fact the boundary connected sum of two rational homology ball fillings of $\eta_j$ and $\eta_{j+2}$,
by Eliashberg [Eli90]. This implies that, up to conjugation, $\eta_j = \xi_j$ and $\eta_{j+2} = \xi_{j+2}$, and
that they both have a unique rational homology ball symplectic filling; again, by [Eli90],
so does $\xi'_j$.

6.4 Isotopy classification

THEOREM 6.6. In $\mathbb{CP}^2$, every symplectic rational unicuspidal curve whose unique singularity is the cone on a torus knot is symplectically isotopic to a complex curve.

(i) The only minimal symplectic embedding of a rational unicuspidal curve with a $(p, p+1)$-singularity with normal Euler number $(p+1)^2$ into a closed symplectic manifold is the unique embedding into $\mathbb{CP}^2$.

(ii) When $p > 2$, there are exactly two minimal symplectic embeddings of a rational unicuspidal curve with a $(p, 4p - 1)$-singularity with normal Euler number $4p^2$ into closed symplectic manifolds. One into $\mathbb{CP}^2$ and another into $S^2 \times S^2$, each unique up to symplectomorphism and symplectic deformation. When $p = 2$, in addition to these two minimal symplectic embeddings into $\mathbb{CP}^2$ and $S^2 \times S^2$, there is also one into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$.

(iii) There are exactly three minimal symplectic embeddings of a rational unicuspidal curve with a (3,22)-singularity with normal Euler number 64. They are embeddings into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2 \# 6 \overline{\mathbb{CP}}^2$ with each unique up to symplectomorphism and symplectic deformation.

(iv) There are exactly six minimal symplectic embeddings of a rational unicuspidal curve with a (6,43)-singularity with normal Euler number 256. There is a single symplectic embedding into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, $\mathbb{CP}^2 \# 4 \overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2 \# 9 \overline{\mathbb{CP}}^2$, up to symplectomorphism. There are two symplectic embeddings into $\mathbb{CP}^2 \# 6 \overline{\mathbb{CP}}^2$, up to symplectomorphism and symplectic deformation.

(v) For the rational unicuspidal curves in the Fibonacci families, there is a unique symplectic embedding in $\mathbb{CP}^2$, and there is always at least one other minimal symplectic embedding into another rational surface with larger $b_2$.

In each of these cases, each symplectic isotopy class of embeddings corresponds to a distinct
minimal symplectic filling of the associated contact manifold, distinguished by their second
homology and intersection forms.

Proof. We start with the non-Fibonacci families.

Suppose we have a symplectic embedding of the cuspidal curve into $(M, \omega)$. Then we have
an embedding of the resolution $\mathcal{A}_p, \mathcal{B}_p, \mathcal{E}_3$, or $\mathcal{E}_6$ into $M \# K \overline{\mathbb{CP}}^2$. By Theorem 3.6 we can iden-
tify $M \# K \overline{\mathbb{CP}}^2$ with $\mathbb{CP}^2 \# N \overline{\mathbb{CP}}^2$ and the (+1)-sphere with a line. In particular, $M$ is either
$\mathbb{CP}^2 \# m \overline{\mathbb{CP}}^2$ for some $m \ge 0$ or $S^2 \times S^2$ (the symplectic structures on these manifolds are unique
up to symplectomorphism and symplectic deformation). The homological embeddings of the res-
olutions in $\mathbb{CP}^2 \# N \overline{\mathbb{CP}}^2$ are classified in Lemmas 6.2–6.5. Next use Lemma 3.5 to find exceptional
spheres representing the classes $e_i$ to blow down to $\mathbb{CP}^2$, keeping track of the intersections of
these exceptional spheres with the resolution to determine how it descends. The core spheres of
$\mathcal{A}_p$ or $\mathcal{B}_p$ descend to two symplectic spheres each in the homology class $h$ of the line. There-
fore, any embedding of a cuspidal curve of type $\mathcal{A}_p$ or $\mathcal{B}_p$ into a closed symplectic manifold
$(M, \omega)$ has a birational derivation to $(\mathbb{CP}^2, L_1 \cup L_2)$ where $L_1$ and $L_2$ are symplectic lines. There
is a unique symplectic isotopy class of two symplectic lines by Proposition 5.1. Therefore, by
Proposition 3.18, there is a unique symplectic isotopy class of such curves for each possible $(M, \omega)$
that they embed into.
in fact the boundary connected sum of two rational homology ball fillings of $\eta_j$ and $\eta_{j+2}$,
by Eliashberg [Eli90]. This implies that, up to conjugation, $\eta_j = \xi_j$ and $\eta_{j+2} = \xi_{j+2}$, and
that they both have a unique rational homology ball symplectic filling; again, by [Eli90],
so does $\xi'_j$.

6.4 Isotopy classification

THEOREM 6.6. In $\mathbb{CP}^2$, every symplectic rational unicuspidal curve whose unique singularity is the cone on a torus knot is symplectically isotopic to a complex curve.

(i) The only minimal symplectic embedding of a rational unicuspidal curve with a $(p, p+1)$-singularity with normal Euler number $(p+1)^2$ into a closed symplectic manifold is the unique embedding into $\mathbb{CP}^2$.

(ii) When $p > 2$, there are exactly two minimal symplectic embeddings of a rational unicuspidal curve with a $(p, 4p - 1)$-singularity with normal Euler number $4p^2$ into closed symplectic manifolds. One into $\mathbb{CP}^2$ and another into $S^2 \times S^2$, each unique up to symplectomorphism and symplectic deformation. When $p = 2$, in addition to these two minimal symplectic embeddings into $\mathbb{CP}^2$ and $S^2 \times S^2$, there is also one into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$.

(iii) There are exactly three minimal symplectic embeddings of a rational unicuspidal curve with a (3,22)-singularity with normal Euler number 64. They are embeddings into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2 \# 6 \overline{\mathbb{CP}}^2$ with each unique up to symplectomorphism and symplectic deformation.

(iv) There are exactly six minimal symplectic embeddings of a rational unicuspidal curve with a (6,43)-singularity with normal Euler number 256. There is a single symplectic embedding into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, $\mathbb{CP}^2 \# 4 \overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2 \# 9 \overline{\mathbb{CP}}^2$, up to symplectomorphism. There are two symplectic embeddings into $\mathbb{CP}^2 \# 6 \overline{\mathbb{CP}}^2$, up to symplectomorphism and symplectic deformation.

(v) For the rational unicuspidal curves in the Fibonacci families, there is a unique symplectic embedding in $\mathbb{CP}^2$, and there is always at least one other minimal symplectic embedding into another rational surface with larger $b_2$.

In each of these cases, each symplectic isotopy class of embeddings corresponds to a distinct
minimal symplectic filling of the associated contact manifold, distinguished by their second
homology and intersection forms.

Proof. We start with the non-Fibonacci families.

Suppose we have a symplectic embedding of the cuspidal curve into $(M, \omega)$. Then we have
an embedding of the resolution $\mathcal{A}_p, \mathcal{B}_p, \mathcal{E}_3$, or $\mathcal{E}_6$ into $M \# K \overline{\mathbb{CP}}^2$. By Theorem 3.6 we can iden-
tify $M \# K \overline{\mathbb{CP}}^2$ with $\mathbb{CP}^2 \# N \overline{\mathbb{CP}}^2$ and the (+1)-sphere with a line. In particular, $M$ is either
$\mathbb{CP}^2 \# m \overline{\mathbb{CP}}^2$ for some $m \ge 0$ or $S^2 \times S^2$ (the symplectic structures on these manifolds are unique
up to symplectomorphism and symplectic deformation). The homological embeddings of the res-
olutions in $\mathbb{CP}^2 \# N \overline{\mathbb{CP}}^2$ are classified in Lemmas 6.2–6.5. Next use Lemma 3.5 to find exceptional
spheres representing the classes $e_i$ to blow down to $\mathbb{CP}^2$, keeping track of the intersections of
these exceptional spheres with the resolution to determine how it descends. The core spheres of
$\mathcal{A}_p$ or $\mathcal{B}_p$ descend to two symplectic spheres each in the homology class $h$ of the line. There-
fore, any embedding of a cuspidal curve of type $\mathcal{A}_p$ or $\mathcal{B}_p$ into a closed symplectic manifold
$(M, \omega)$ has a birational derivation to $(\mathbb{CP}^2, L_1 \cup L_2)$ where $L_1$ and $L_2$ are symplectic lines. There
is a unique symplectic isotopy class of two symplectic lines by Proposition 5.1. Therefore, by
Proposition 3.18, there is a unique symplectic isotopy class of such curves for each possible $(M, \omega)$
that they embed into.
The configuration $\mathcal{A}_p$. Lemma 6.2 gives a unique homological embedding of $\mathcal{A}_p$ (which is obtained from the cuspidal curve by blowing up $2p+1$ times) into $\mathbb{CP}^2\#(2p+1)\overline{\mathbb{CP}}^2$. Therefore, the only relatively minimal embedding of these cuspidal curves is into $\mathbb{CP}^2$.

The configuration $\mathcal{B}_p$. Lemma 6.3 gives two homological embeddings of $\mathcal{B}_p$ (which is obtained from the cuspidal curve by blowing up $2p+2$ times), one into $\mathbb{CP}^2\#(2p+3)\overline{\mathbb{CP}}^2$ and the other into $\mathbb{CP}^2\#(2p+2)\overline{\mathbb{CP}}^2$, plus an additional embedding when $p=2$ into $\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2$. Therefore the cuspidal curve of type $\mathcal{B}_p$ has one symplectic embedding into $\mathbb{CP}^2$ and one (two if $p=2$) more into a manifold $X$ which is either $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$. We now see how to determine whether the manifold is $S^2 \times S^2$ or $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ in this latter case. Here $C$ is the rational cuspidal curve in $X$, so using our chosen resolution, the total transform of $C$ embeds in a blow-up $\bar{X}$ of $X$. The intersection form of $X$ is recovered from that of $\bar{X}$ by 'algebraically blowing down' all the exceptional divisors in the total transform of $C$. This boils down to taking the orthogonal of the homology classes. For the homological embedding from Lemma 6.3 which applies to all $p \ge 2$ cases, we need the orthogonal of the classes

$$h - e_0 - e_1, \quad e_{2p-2} - e_{2p-1} - e_{2p}, \\ e_0 - e_1 - \cdots - e_{p-1}, \quad e_i - e_{i+1} \quad (i \in \{1, \dots, 2p-3, 2p, 2p+1\})$$

in $H_2(\bar{X}) = \langle h, e_0, \dots, e_{2p+2} \rangle$. Orthogonality to the classes $e_i - e_{i+1}$ forces classes in the orthogonal to be of the form

$$ah - b_0e_0 - b_1(e_1 + \cdots + e_{2p-2}) - b_{2p-1}e_{2p-1} - b_{2p}(e_{2p} + e_{2p+1} + e_{2p+2}).$$

The orthogonal is generated by $C_1 = ph - (p-1)e_0 - e_1 - \cdots - e_{2p-1}$, $C_2 = ph - (p-1)e_0 - e_1 - \cdots - e_{2p-2} - e_{2p} - e_{2p+1} - e_{2p+2}$. We calculate $C_1^2 = 0$ and that $C_2^2 = -2$, so that the intersection form of $X$ is even, hence necessarily $X = S^2 \times S^2$. As another way that we could have seen this, we show in Proposition 6.7 that the latter embedding is obtained from the former by doing a rational blow-up of $T^*\mathbb{RP}^2$, thus showing that the ambient 4-manifold is indeed $S^2 \times S^2$.

For the additional homological embedding in the $p=2$ case, we claim this corresponds to an embedding of $C$ into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. In this case, the intersection form of $X$ is isometric to the subspace $\langle h - e_0 - e_1, e_1 - e_2, e_0 - e_1 - e_3, e_3 - e_4, e_4 - e_5, e_2 - e_6 \rangle^\perp \subset H_2(\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2)$. This subspace is generated by the classes $h - e_0 - e_3 - e_4 - e_5$ and $2h - e_0 - e_1 - e_2 - e_6$. As the self-intersection of the first class is $-3$, which is odd, $\bar{X}$ is $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$.

The configuration $\mathcal{E}_3$. For $\mathcal{E}_3$, after blowing down the exceptional spheres in the $e_i$ classes using Lemma 3.5, the configuration descends to a symplectic nodal or cuspidal cubic (depending on whether the exceptional sphere of class $e_0$ intersects the configuration transversally or tangentially), together with a symplectic line which is tangent to the cubic with multiplicity 3. Each such configuration has a unique symplectic isotopy class by Proposition 5.12 with $d=3$. To connect these two possibilities, observe that each of these possibilities is realizable in the complex setting. There is a deformation from a complex algebraic cuspidal cubic $\{x^3 - y^2z = 0\}$ with inflection line $\{z=0\}$, to a nodal cubic $\{x^3 - y^2z - \varepsilon x^2z = 0\}$ with the same inflection line $\{z=0\}$. Blowing up at the cusp or node in the $\varepsilon$ family provides an equisingular isotopy of the proper transforms in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. Therefore, there is a unique symplectic isotopy class of the configuration $C'$ in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ consisting of one smooth rational curve in the class $3h - 2e_0$ and a line in class $h$, such that the two components intersect tangentially at a single point of multiplicity three. The birational transformation described previously shows that the configuration $\mathcal{E}_3$ has a birational transformation to $C'$ in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. Therefore, by Proposition 3.18, for each possible $(M, \omega)$ determined by a homological embedding, there is
The configuration $\mathcal{A}_p$. Lemma 6.2 gives a unique homological embedding of $\mathcal{A}_p$ (which is obtained from the cuspidal curve by blowing up $2p+1$ times) into $\mathbb{CP}^2\#(2p+1)\overline{\mathbb{CP}}^2$. Therefore, the only relatively minimal embedding of these cuspidal curves is into $\mathbb{CP}^2$.

The configuration $\mathcal{B}_p$. Lemma 6.3 gives two homological embeddings of $\mathcal{B}_p$ (which is obtained from the cuspidal curve by blowing up $2p+2$ times), one into $\mathbb{CP}^2\#(2p+3)\overline{\mathbb{CP}}^2$ and the other into $\mathbb{CP}^2\#(2p+2)\overline{\mathbb{CP}}^2$, plus an additional embedding when $p=2$ into $\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2$. Therefore the cuspidal curve of type $\mathcal{B}_p$ has one symplectic embedding into $\mathbb{CP}^2$ and one (two if $p=2$) more into a manifold $X$ which is either $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$. We now see how to determine whether the manifold is $S^2 \times S^2$ or $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ in this latter case. Here $C$ is the rational cuspidal curve in $X$, so using our chosen resolution, the total transform of $C$ embeds in a blow-up $\overline{X}$ of $X$. The intersection form of $X$ is recovered from that of $\overline{X}$ by 'algebraically blowing down' all the exceptional divisors in the total transform of $C$. This boils down to taking the orthogonal of the homology classes. For the homological embedding from Lemma 6.3 which applies to all $p \ge 2$ cases, we need the orthogonal of the classes

$$h - e_0 - e_1, \quad e_{2p-2} - e_{2p-1} - e_{2p}, \\ e_0 - e_1 - \cdots - e_{p-1}, \quad e_i - e_{i+1} \quad (i \in \{1, \dots, 2p-3, 2p, 2p+1\})$$

in $H_2(\overline{X}) = \langle h, e_0, \dots, e_{2p+2} \rangle$. Orthogonality to the classes $e_i - e_{i+1}$ forces classes in the orthogonal to be of the form

$$ah - b_0e_0 - b_1(e_1 + \cdots + e_{2p-2}) - b_{2p-1}e_{2p-1} - b_{2p}(e_{2p} + e_{2p+1} + e_{2p+2}).$$

The orthogonal is generated by $C_1 = ph - (p-1)e_0 - e_1 - \cdots - e_{2p-1}$, $C_2 = ph - (p-1)e_0 - e_1 - \cdots - e_{2p-2} - e_{2p} - e_{2p+1} - e_{2p+2}$. We calculate $C_1^2 = 0$ and that $C_2^2 = -2$, so that the intersection form of $X$ is even, hence necessarily $X = S^2 \times S^2$. As another way that we could have seen this, we show in Proposition 6.7 that the latter embedding is obtained from the former by doing a rational blow-up of $T^*\mathbb{RP}^2$, thus showing that the ambient 4-manifold is indeed $S^2 \times S^2$.

For the additional homological embedding in the $p=2$ case, we claim this corresponds to an embedding of $C$ into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. In this case, the intersection form of $X$ is isometric to the subspace $\langle h - e_0 - e_1, e_1 - e_2, e_0 - e_1 - e_3, e_3 - e_4, e_4 - e_5, e_2 - e_6 \rangle^\perp \subset H_2(\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2)$. This subspace is generated by the classes $h - e_0 - e_3 - e_4 - e_5$ and $2h - e_0 - e_1 - e_2 - e_6$. As the self-intersection of the first class is $-3$, which is odd, $\overline{X}$ is $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$.

The configuration $\mathcal{E}_3$. For $\mathcal{E}_3$, after blowing down the exceptional spheres in the $e_i$ classes using Lemma 3.5, the configuration descends to a symplectic nodal or cuspidal cubic (depending on whether the exceptional sphere of class $e_0$ intersects the configuration transversally or tangentially), together with a symplectic line which is tangent to the cubic with multiplicity 3. Each such configuration has a unique symplectic isotopy class by Proposition 5.12 with $d=3$. To connect these two possibilities, observe that each of these possibilities is realizable in the complex setting. There is a deformation from a complex algebraic cuspidal cubic $\{x^3 - y^2z = 0\}$ with inflection line $\{z=0\}$, to a nodal cubic $\{x^3 - y^2z - \varepsilon x^2z = 0\}$ with the same inflection line $\{z=0\}$. Blowing up at the cusp or node in the $\varepsilon$ family provides an equisingular isotopy of the proper transforms in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. Therefore, there is a unique symplectic isotopy class of the configuration $C'$ in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ consisting of one smooth rational curve in the class $3h - 2e_0$ and a line in class $h$, such that the two components intersect tangentially at a single point of multiplicity three. The birational transformation described previously shows that the configuration $\mathcal{E}_3$ has a birational transformation to $C'$ in $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$. Therefore, by Proposition 3.18, for each possible $(M, \omega)$ determined by a homological embedding, there is
a unique symplectic isotopy class of embeddings of the cuspidal curve of type $\mathcal{E}_3$. The three homological embeddings of $\mathcal{E}_3$ (which is obtained from the cuspidal curve by seven blow-ups) are into $\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2$, $\mathbb{CP}^2\#8\overline{\mathbb{CP}}^2$ and $\mathbb{CP}^2\#13\overline{\mathbb{CP}}^2$. Therefore, the cuspidal curve has one minimal symplectic embedding in $\mathbb{CP}^2$, one in either $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$, and one into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$. To determine that the second embedding is into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ instead of $S^2 \times S^2$ we find a homology basis complementary to the classes of the exceptional divisors in the embedding of $\mathcal{E}_3$ ($3h - 2e_0 - e_1 - \cdots - e_6$, $e_1 - e_7$, $e_2 - e_1$, $e_3 - e_2$, $e_4 - e_3$, $e_5 - e_4$, and $e_6 - e_5$). Such a basis is given by $10h - 6e_0 - 3(e_1 + \cdots + e_7)$ and $6h - 3e_0 - 2(e_1 + \cdots + e_7)$. The intersection form generated by this basis is the odd form for $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$:

$$ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. $$

The configuration $\mathcal{E}_6$. By Lemma 6.5, there are six different homological embeddings of $\mathcal{E}_6$ into blow-ups of $\mathbb{CP}^2$. We show that each such homological embedding has a unique symplectic isotopy class. In each case, we blow down exceptional spheres until we reach $\mathbb{CP}^2$ using Lemma 3.5, and consider the image of $\mathcal{E}_6$.

For either of the homological embeddings where the upper chain is described by (6.5), the image of $\mathcal{E}_6$ in $\mathbb{CP}^2$ is a nodal or cuspidal cubic $R$ (depending on whether the exceptional sphere representing $e_0$ intersects the component representing $3h - 2e_0 - e_1 - \cdots - e_9$ transversally or tangentially), together with an inflection line $T$ tangent to the cubic with multiplicity three, and another line $L$ which passes through the singularity of $R$ and through the point $R \cap T$ (intersecting the other components transversally at each of these points). The configuration of $R \cup T$ has a unique symplectic isotopy class by Proposition 5.12, so the configuration $R \cup T \cup L$ has a unique symplectic isotopy class by Proposition 5.1. Therefore, any two embeddings of $\mathcal{E}_6$ which descend to the version of this configuration where $R$ is nodal are symplectically isotopic, and similarly any two which descend to the version of this configuration where $R$ is cuspidal are symplectically isotopic. We connect the nodal and cuspidal cases of $R \cup T$ as in the $\mathcal{E}_3$ case, and augment the deformation from the cusp to the node by the line $L$ to connect the nodal and cuspidal instances of $R \cup T \cup L$.

For the homological embedding where the upper chain is described by (6.4), the embedding of $\mathcal{E}_6$ descends under blow downs to a configuration $R \cup T \cup L'$. Again $R$ is a nodal or cuspidal cubic, and $T$ is again an inflection line (tangent of order three to $C$). This time $L'$ is a line which passes transversally through $R \cap T$ and intersects $R$ again transversally at two smooth points (the first point is the image of exceptional spheres representing $e_6$, $e_7$, and $e_8$ and the second point is the image of the exceptional sphere representing $e_9$). Here $R \cup T$ has a unique symplectic isotopy class by Proposition 5.12, and we can add $L'$ maintaining a unique symplectic isotopy class using Proposition 5.1. The nodal and cuspidal cases are connected as in the $\mathcal{E}_3$ case.

For the homological embeddings where the upper chain is described by (6.6), the embedding of $\mathcal{E}_6$ descends to $R \cup T \cup L''$ where $R$ and $T$ are the same as the previous cases and $L''$ passes transversally through $R \cap T$ and intersects $R$ at one other smooth point with a simple tangency (the image of the exceptional spheres representing $e_7$, $e_8$, and $e_9$). In this case, we show that there is a unique symplectic isotopy class of such a configuration when the cubic is nodal and no realization of the configuration if the cubic has a cusp.

To prove this, choose an almost complex structure $J$ which makes the cubic $R$ and its inflection line $T$ $J$-holomorphic (note the space of such $J$ is contractible by Lemma 3.4). Let $p = R \cap T$. Consider the pencil of $J$-lines through $p$, $\pi: \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$ and restrict this
a unique symplectic isotopy class of embeddings of the cuspidal curve of type $\mathcal{E}_3$. The three homological embeddings of $\mathcal{E}_3$ (which is obtained from the cuspidal curve by seven blow-ups) are into $\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2$, $\mathbb{CP}^2\#8\overline{\mathbb{CP}}^2$ and $\mathbb{CP}^2\#13\overline{\mathbb{CP}}^2$. Therefore, the cuspidal curve has one minimal symplectic embedding in $\mathbb{CP}^2$, one in either $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$, and one into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$. To determine that the second embedding is into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ instead of $S^2 \times S^2$ we find a homology basis complementary to the classes of the exceptional divisors in the embedding of $\mathcal{E}_3$ ($3h - 2e_0 - e_1 - \cdots - e_6$, $e_1 - e_7$, $e_2 - e_1$, $e_3 - e_2$, $e_4 - e_3$, $e_5 - e_4$, and $e_6 - e_5$). Such a basis is given by $10h - 6e_0 - 3(e_1 + \cdots + e_7)$ and $6h - 3e_0 - 2(e_1 + \cdots + e_7)$. The intersection form generated by this basis is the odd form for $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$:

$$ \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. $$

The configuration $\mathcal{E}_6$. By Lemma 6.5, there are six different homological embeddings of $\mathcal{E}_6$ into blow-ups of $\mathbb{CP}^2$. We show that each such homological embedding has a unique symplectic isotopy class. In each case, we blow down exceptional spheres until we reach $\mathbb{CP}^2$ using Lemma 3.5, and consider the image of $\mathcal{E}_6$.

For either of the homological embeddings where the upper chain is described by (6.5), the image of $\mathcal{E}_6$ in $\mathbb{CP}^2$ is a nodal or cuspidal cubic $R$ (depending on whether the exceptional sphere representing $e_0$ intersects the component representing $3h - 2e_0 - e_1 - \cdots - e_9$ transversally or tangentially), together with an inflection line $T$ tangent to the cubic with multiplicity three, and another line $L$ which passes through the singularity of $R$ and through the point $R \cap T$ (intersecting the other components transversally at each of these points). The configuration of $R \cup T$ has a unique symplectic isotopy class by Proposition 5.12, so the configuration $R \cup T \cup L$ has a unique symplectic isotopy class by Proposition 5.1. Therefore, any two embeddings of $\mathcal{E}_6$ which descend to the version of this configuration where $R$ is nodal are symplectically isotopic, and similarly any two which descend to the version of this configuration where $R$ is cuspidal are symplectically isotopic. We connect the nodal and cuspidal cases of $R \cup T$ as in the $\mathcal{E}_3$ case, and augment the deformation from the cusp to the node by the line $L$ to connect the nodal and cuspidal instances of $R \cup T \cup L$.

For the homological embedding where the upper chain is described by (6.4), the embedding of $\mathcal{E}_6$ descends under blow downs to a configuration $R \cup T \cup L'$. Again $R$ is a nodal or cuspidal cubic, and $T$ is again an inflection line (tangent of order three to $C$). This time $L'$ is a line which passes transversally through $R \cap T$ and intersects $R$ again transversally at two smooth points (the first point is the image of exceptional spheres representing $e_6$, $e_7$, and $e_8$ and the second point is the image of the exceptional sphere representing $e_9$). Here $R \cup T$ has a unique symplectic isotopy class by Proposition 5.12, and we can add $L'$ maintaining a unique symplectic isotopy class using Proposition 5.1. The nodal and cuspidal cases are connected as in the $\mathcal{E}_3$ case.

For the homological embeddings where the upper chain is described by (6.6), the embedding of $\mathcal{E}_6$ descends to $R \cup T \cup L''$ where $R$ and $T$ are the same as the previous cases and $L''$ passes transversally through $R \cap T$ and intersects $R$ at one other smooth point with a simple tangency (the image of the exceptional spheres representing $e_7$, $e_8$, and $e_9$). In this case, we show that there is a unique symplectic isotopy class of such a configuration when the cubic is nodal and no realization of the configuration if the cubic has a cusp.

To prove this, choose an almost complex structure $J$ which makes the cubic $R$ and its inflection line $T$ $J$-holomorphic (note the space of such $J$ is contractible by Lemma 3.4). Let $p = R \cap T$. Consider the pencil of $J$-lines through $p$, $\pi : \mathbb{CP}^2 \setminus \{p\} \to \mathbb{CP}^1$ and restrict this
to $R \setminus \{p\}$. This restriction extends over $p$ by sending $p$ to the image of the line tangent to $R$ at $p$. We pre-compose with the normalization of $R$.

In the cusp case, projecting the cubic from the inflection point gives a degree-two map $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ with at least two ramification points (corresponding to the inflection line and the cusp, respectively). Therefore, Riemann-Hurwitz reads: $2 = 2 \cdot 2 - \sum (e_\pi(p) - 1)$, which implies that these are the only two ramification points, from which we deduce that there is no other tangent drawn to the cubic from the inflection point. In the nodal case, the projection $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ of the cubic from $p$ has no ramification at the node, and ramification two at the inflection line, so there is exactly another point of ramification two, which corresponds to a tangent line to the cubic.

Therefore, for each fixed homological embedding of the resolution, there is a unique symplectic embedding of the cuspidal curve up to isotopy. The resolution is obtained from the cuspidal curve by performing 10 blow-ups. Therefore, the embeddings of the resolution into $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$, $\mathbb{CP}^2\#11\overline{\mathbb{CP}}^2$, two into $\mathbb{CP}^2\#16\overline{\mathbb{CP}}^2$, one into $\mathbb{CP}^2\#19\overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2\#14\overline{\mathbb{CP}}^2$ correspond to embeddings of the cuspidal curve into $\mathbb{CP}^2$, $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$, two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$, and one into $\mathbb{CP}^2\#9\overline{\mathbb{CP}}^2$ and $\mathbb{CP}^2\#4\overline{\mathbb{CP}}^2$.

It remains to distinguish the two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$ and prove that this cuspidal curve embeds into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ instead of $S^2 \times S^2$. For the latter question, the answer is similar to the $\mathcal{E}_3$ case, but $e_7$ is replaced by $e_{10}$. Therefore, a basis for the homology complementary to the exceptional divisors is given by $10h - 6e_0 - 3(e_1 + \cdots + e_6 + e_{10})$ and $6h - 3e_0 - 2(e_1 + \cdots + e_6 + e_{10})$ which have the odd intersection form of $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$.

To distinguish the two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$, we show that their complementary symplectic fillings have different intersection forms. The two homological embeddings we are considering from Lemma 6.5 both use option (i) for the lower chain, and differ in the upper chain, being either option (6.4) or (6.6).

A homology basis for the complement of the embedding of the cap (not just the exceptional spheres but also the proper transform representing $h$) in the first case (option (6.4)) is given by the following:

$$ \{e_0 - e_4 - e_5, e_2 - e_1 - e_{10} - \cdots - e_{15}, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_0 + e_9 - e_6 - e_7 - e_8\} $$

giving an intersection form represented by

$$ Q_1 = \begin{bmatrix} -3 & 0 & 0 & -1 & 0 & -1 \\ 0 & -8 & -1 & 0 & 0 & 0 \\ 0 & -1 & -2 & 1 & 0 & 0 \\ -1 & 0 & 1 & -2 & 1 & 0 \\ 0 & 0 & 0 & 1 & -2 & 0 \\ -1 & 0 & 0 & 0 & 0 & -5 \end{bmatrix}. $$

In the second case (option (6.6)) a basis is given by the following:

$$ \{e_2 - e_1 - e_{10} - \cdots - e_{15}, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6, e_0 - e_5 - e_6\} $$
to $R \setminus \{p\}$. This restriction extends over $p$ by sending $p$ to the image of the line tangent to $R$ at $p$. We pre-compose with the normalization of $R$.

In the cusp case, projecting the cubic from the inflection point gives a degree-two map $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ with at least two ramification points (corresponding to the inflection line and the cusp, respectively). Therefore, Riemann-Hurwitz reads: $2 = 2 \cdot 2 - \sum (e_\pi(p) - 1)$, which implies that these are the only two ramification points, from which we deduce that there is no other tangent drawn to the cubic from the inflection point. In the nodal case, the projection $\pi: \mathbb{CP}^1 \to \mathbb{CP}^1$ of the cubic from $p$ has no ramification at the node, and ramification two at the inflection line, so there is exactly another point of ramification two, which corresponds to a tangent line to the cubic.

Therefore, for each fixed homological embedding of the resolution, there is a unique symplectic embedding of the cuspidal curve up to isotopy. The resolution is obtained from the cuspidal curve by performing 10 blow-ups. Therefore, the embeddings of the resolution into $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$, $\mathbb{CP}^2\#11\overline{\mathbb{CP}}^2$, two into $\mathbb{CP}^2\#16\overline{\mathbb{CP}}^2$, one into $\mathbb{CP}^2\#19\overline{\mathbb{CP}}^2$, and $\mathbb{CP}^2\#14\overline{\mathbb{CP}}^2$ correspond to embeddings of the cuspidal curve into $\mathbb{CP}^2$, $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ or $S^2 \times S^2$, two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$, and one into $\mathbb{CP}^2\#9\overline{\mathbb{CP}}^2$ and $\mathbb{CP}^2\#4\overline{\mathbb{CP}}^2$.

It remains to distinguish the two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$ and prove that this cuspidal curve embeds into $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$ instead of $S^2 \times S^2$. For the latter question, the answer is similar to the $\mathcal{E}_3$ case, but $e_7$ is replaced by $e_{10}$. Therefore, a basis for the homology complementary to the exceptional divisors is given by $10h - 6e_0 - 3(e_1 + \dots + e_6 + e_{10})$ and $6h - 3e_0 - 2(e_1 + \dots + e_6 + e_{10})$ which have the odd intersection form of $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$.

To distinguish the two embeddings into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$, we show that their complementary symplectic fillings have different intersection forms. The two homological embeddings we are considering from Lemma 6.5 both use option (i) for the lower chain, and differ in the upper chain, being either option (6.4) or (6.6).

A homology basis for the complement of the embedding of the cap (not just the exceptional spheres but also the proper transform representing $h$) in the first case (option (6.4)) is given by the following:

$$ \{e_0 - e_4 - e_5, e_2 - e_1 - e_{10} - \dots - e_{15}, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_0 + e_9 - e_6 - e_7 - e_8\} $$

giving an intersection form represented by

$$ Q_1 = \begin{bmatrix} -3 & 0 & 0 & -1 & 0 & -1 \\ 0 & -8 & -1 & 0 & 0 & 0 \\ 0 & -1 & -2 & 1 & 0 & 0 \\ -1 & 0 & 1 & -2 & 1 & 0 \\ 0 & 0 & 0 & 1 & -2 & 0 \\ -1 & 0 & 0 & 0 & 0 & -5 \end{bmatrix}. $$

In the second case (option (6.6)) a basis is given by the following:

$$ \{e_2 - e_1 - e_{10} - \dots - e_{15}, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6, e_0 - e_5 - e_6\} $$
giving an intersection form represented by

$$Q_2 = \begin{bmatrix} -8 & -1 & 0 & 0 & 0 & 0 \\ -1 & -2 & 1 & 0 & 0 & 0 \\ 0 & 1 & -2 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & -1 & 0 & -3 \end{bmatrix}$$

As det $Q_1 = 256$ and det $Q_2 = 64$, the two embeddings are not even homeomorphic.

The Fibonacci families. In the previous subsection, we have shown that in each case the cuspidal contact structure has a unique rational homology ball filling up to symplectic deformation. In particular, there is a unique symplectomorphism class of curves in $\mathbb{CP}^2$ for each of the members of either family, and by Gromov there is a unique symplectic isotopy class, so the complex representatives are necessarily in that isotopy class. In addition, we observe that each member in the first Fibonacci family has at least one other filling, whereas each member in the second Fibonacci family has at least two. Indeed, in the first Fibonacci family, each of the cuspidal contact 3-manifolds ($Y_C, \xi_C$) is a universally tight lens space. Each universally tight contact lens space admits at least another filling (namely, a plumbing of symplectic spheres); gluing this filling to the cuspidal cap yields a closed symplectic 4-manifold into which $C$ embeds (with minimal complement). In the second Fibonacci family, each of the cuspidal contact 3-manifolds ($Y_C, \xi_C$) is a connected sum of two distinct universally tight lens spaces (each admitting a rational homology ball symplectic filling), so it has at least three non-rational homology ball minimal symplectic fillings, obtained by boundary connected summing either a plumbing filling and a rational homology ball filling for each of the summand, or the two plumbings. To show that these fillings are distinct, we look at the order of their first homology: the two rational homology balls have $H_1 = \mathbb{Z}/F_j\mathbb{Z}$ and $H_1 = \mathbb{Z}/F_{j+2}\mathbb{Z}$, respectively, whereas the plumbings are simply connected, so the four fillings we exhibited have $|H_1| = F_jF_{j+2}, F_j, F_{j+2}$, and 1, respectively. $\square$

## 6.5 Rational blow-down relations

In this section we study the relationships between the two fillings of the cuspidal contact structure associated with the rational unicuspidal curves in the second family, i.e. those with a singularity of type $(p, 4p - 1)$. The two fillings are complements of concave neighborhoods of the two different embeddings.

We study the two sporadic cases $(3, 22)$ and $(6, 43)$ in a companion paper [GS21], by showing that the corresponding cuspidal contact structure is, in fact, the canonical contact structure on the link of a singularity (compare with Stipsicz, Szabó, and Wahl [SSzW08]) and studying the relationship between the different fillings.

As argued in §6.3, the two Fibonacci families correspond to lens spaces with their canonical contact structure, for which fillings and rational blow-down relationships have been studied extensively [BO18].

Recall that when $p > 2$ each rational curve with self-intersection $4p^2$ and a singularity of type $(p, 4p - 1)$ comes with two fillings, $W_0, W_1$, with $b_2(W_i) = i$. (See Remark 6.8 for the $p = 2$ case.)

**PROPOSITION 6.7.** Let $C$ be a rational unicuspidal curve with a singularity of type $(p, 4p - 1)$ and self-intersection $4p^2$, and $\xi$ the corresponding cuspidal contact structure. Then two of the symplectic fillings of $\xi$ are related by a rational blow-down of a symplectic $(-4)$-sphere.
giving an intersection form represented by

$$Q_2 = \begin{bmatrix} -8 & -1 & 0 & 0 & 0 & 0 \\ -1 & -2 & 1 & 0 & 0 & 0 \\ 0 & 1 & -2 & 1 & 0 & 0 \\ 0 & 0 & 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & -1 & 0 & -3 \end{bmatrix}$$

As det $Q_1 = 256$ and det $Q_2 = 64$, the two embeddings are not even homeomorphic.

The Fibonacci families. In the previous subsection, we have shown that in each case the cuspidal contact structure has a unique rational homology ball filling up to symplectic deformation. In particular, there is a unique symplectomorphism class of curves in $\mathbb{CP}^2$ for each of the members of either family, and by Gromov there is a unique symplectic isotopy class, so the complex representatives are necessarily in that isotopy class. In addition, we observe that each member in the first Fibonacci family has at least one other filling, whereas each member in the second Fibonacci family has at least two. Indeed, in the first Fibonacci family, each of the cuspidal contact 3-manifolds $(Y_C, \xi_C)$ is a universally tight lens space. Each universally tight contact lens space admits at least another filling (namely, a plumbing of symplectic spheres); gluing this filling to the cuspidal cap yields a closed symplectic 4-manifold into which $C$ embeds (with minimal complement). In the second Fibonacci family, each of the cuspidal contact 3-manifolds $(Y_C, \xi_C)$ is a connected sum of two distinct universally tight lens spaces (each admitting a rational homology ball symplectic filling), so it has at least three non-rational homology ball minimal symplectic fillings, obtained by boundary connected summing either a plumbing filling and a rational homology ball filling for each of the summand, or the two plumbings. To show that these fillings are distinct, we look at the order of their first homology: the two rational homology balls have $H_1 = \mathbb{Z}/F_j\mathbb{Z}$ and $H_1 = \mathbb{Z}/F_{j+2}\mathbb{Z}$, respectively, whereas the plumbings are simply connected, so the four fillings we exhibited have $|H_1| = F_jF_{j+2}, F_j, F_{j+2}$, and 1, respectively. $\square$

## 6.5 Rational blow-down relations

In this section we study the relationships between the two fillings of the cuspidal contact structure associated with the rational unicuspidal curves in the second family, i.e. those with a singularity of type $(p, 4p - 1)$. The two fillings are complements of concave neighborhoods of the two different embeddings.

We study the two sporadic cases $(3, 22)$ and $(6, 43)$ in a companion paper [GS21], by showing that the corresponding cuspidal contact structure is, in fact, the canonical contact structure on the link of a singularity (compare with Stipsicz, Szabó, and Wahl [SSzW08]) and studying the relationship between the different fillings.

As argued in §6.3, the two Fibonacci families correspond to lens spaces with their canonical contact structure, for which fillings and rational blow-down relationships have been studied extensively [BO18].

Recall that when $p > 2$ each rational curve with self-intersection $4p^2$ and a singularity of type $(p, 4p - 1)$ comes with two fillings, $W_0, W_1$, with $b_2(W_i) = i$. (See Remark 6.8 for the $p = 2$ case.)

**PROPOSITION 6.7.** Let $C$ be a rational unicuspidal curve with a singularity of type $(p, 4p - 1)$ and self-intersection $4p^2$, and $\xi$ the corresponding cuspidal contact structure. Then two of the symplectic fillings of $\xi$ are related by a rational blow-down of a symplectic $(-4)$-sphere.
*Proof.* We need to look for a symplectic (-4)-sphere in the filling with $b_2 = 1$, which we call $W_1$. Recall from Lemma 6.3 that the homology classes of the embeddings of the spheres of the plumbing $B_p$ whose complement is $W_1$ are

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

The homology class $e_{2p-1} - e_{2p} - e_{2p+1} - e_{2p+2}$ generates the orthogonal of these classes in $H_2(\mathbb{CP}^2\#(2p+2)\overline{\mathbb{CP}}^2)$. We can choose to realize the configuration above by performing the last three blow-ups on the exceptional sphere in class $e_{2p-1}$. Then, the proper transform of $e_{2p-1}$ is an embedded symplectic sphere in the class $e_{2p-1} - e_{2p} - e_{2p+1} - e_{2p+2}$ with self-intersection -4, in the complement of the embedding of $B_p$. As $\xi$ only has two strong symplectic fillings, the rational blow-down of this (-4)-sphere must give the unique rational homology ball filling. $\square$

*Remark 6.8.* The same rational blow-down relation holds in the $p = 2$ case for the two fillings associated with the homological cases that generalize for $p > 2$. The additional filling with $b_2 = 1$ that appears only in the $p = 2$ case is not related to the rational homology ball filling via rational blow-down. This is because the intersection form of this exceptional filling is $\langle -16 \rangle$, so it cannot contain a (-4)-sphere to rationally blow down. To calculate the intersection form, start with the homological embedding of Lemma 6.3 and observe that the orthogonal to the subspace generated by the components of the total transform of $C$ in $H_2(\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2)$ is generated by the class $e_0 - e_1 - e_2 + 2e_3 + 2e_4 + 2e_5 - e_6$, which has self-intersection -16.

# 7. Low-degree cuspidal curves

The goal of this section is to prove Theorem 1.3; namely, we want to prove that every rational cuspidal curve of degree up to five has a unique isotopy class, and that this class contains a complex representative.

We split the proof degree by degree; there are no singular lines or conics, so we only need to start at degree three. By the degree-genus formula (2.1), a cuspidal cubic can only have one singularity, which is necessarily a simple cusp. This case was already considered in §6 and in [O005].

In the next two subsections, we look at quartics and quintics. As in the previous section, we actually provide classifications of symplectic embeddings of these cuspidal curves (with prescribed normal Euler number) into any closed symplectic manifold, equivalently classifying the symplectic fillings of the associated contact structures.

## 7.1 Quartics

By the degree-genus formula (2.1), a cuspidal quartic can only have multiplicity multi-sequence [[3]] or [[2, 2, 2]]; correspondingly, the allowed types of configurations of singularities are the following:

* [3]: a single cusp of type (3, 4) which we showed has a unique symplectic isotopy class of embeddings into $\mathbb{CP}^2$ in §6;

* [2, 2, 2]: a single cusp of type (2, 7) which was also shown to have a unique embedding into $\mathbb{CP}^2$ in §6;

* [2, 2], [2]: one cusp of type (2, 5) and one of type (2, 3) will be shown to have a unique relatively minimal symplectic embedding into each of $\mathbb{CP}^2$, $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$, and $S^2 \times S^2$ in Proposition 7.1;
*Proof.* We need to look for a symplectic $(-4)$-sphere in the filling with $b_2 = 1$, which we call $W_1$. Recall from Lemma 6.3 that the homology classes of the embeddings of the spheres of the plumbing $\mathcal{B}_p$ whose complement is $W_1$ are

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

The homology class $e_{2p-1} - e_{2p} - e_{2p+1} - e_{2p+2}$ generates the orthogonal of these classes in $H_2(\mathbb{CP}^2\#(2p+2)\overline{\mathbb{CP}}^2)$. We can choose to realize the configuration above by performing the last three blow-ups on the exceptional sphere in class $e_{2p-1}$. Then, the proper transform of $e_{2p-1}$ is an embedded symplectic sphere in the class $e_{2p-1} - e_{2p} - e_{2p+1} - e_{2p+2}$ with self-intersection $-4$, in the complement of the embedding of $\mathcal{B}_p$. As $\xi$ only has two strong symplectic fillings, the rational blow-down of this $(-4)$-sphere must give the unique rational homology ball filling. $\square$

*Remark 6.8.* The same rational blow-down relation holds in the $p = 2$ case for the two fillings associated with the homological cases that generalize for $p > 2$. The additional filling with $b_2 = 1$ that appears only in the $p = 2$ case is not related to the rational homology ball filling via rational blow-down. This is because the intersection form of this exceptional filling is $\langle -16 \rangle$, so it cannot contain a $(-4)$-sphere to rationally blow down. To calculate the intersection form, start with the homological embedding of Lemma 6.3 and observe that the orthogonal to the subspace generated by the components of the total transform of $C$ in $H_2(\mathbb{CP}^2\#7\overline{\mathbb{CP}}^2)$ is generated by the class $e_0 - e_1 - e_2 + 2e_3 + 2e_4 + 2e_5 - e_6$, which has self-intersection $-16$.

# 7. Low-degree cuspidal curves

The goal of this section is to prove Theorem 1.3; namely, we want to prove that every rational cuspidal curve of degree up to five has a unique isotopy class, and that this class contains a complex representative.

We split the proof degree by degree; there are no singular lines or conics, so we only need to start at degree three. By the degree-genus formula (2.1), a cuspidal cubic can only have one singularity, which is necessarily a simple cusp. This case was already considered in §6 and in [O005].

In the next two subsections, we look at quartics and quintics. As in the previous section, we actually provide classifications of symplectic embeddings of these cuspidal curves (with prescribed normal Euler number) into any closed symplectic manifold, equivalently classifying the symplectic fillings of the associated contact structures.

## 7.1 Quartics

By the degree-genus formula (2.1), a cuspidal quartic can only have multiplicity multi-sequence [[3]] or [[2, 2, 2]]; correspondingly, the allowed types of configurations of singularities are the following:

* [3]: a single cusp of type (3, 4) which we showed has a unique symplectic isotopy class of embeddings into $\mathbb{CP}^2$ in §6;

* [2, 2, 2]: a single cusp of type (2, 7) which was also shown to have a unique embedding into $\mathbb{CP}^2$ in §6;

* [2, 2], [2]: one cusp of type (2, 5) and one of type (2, 3) will be shown to have a unique relatively minimal symplectic embedding into each of $\mathbb{CP}^2$, $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$, and $S^2 \times S^2$ in Proposition 7.1;
<figure><img src="image_29.png" /><figcaption>FIGURE 22. A resolution of a rational cuspidal quartic with one cusp of type (2,3) and one of type (2,5) with the possible homological embeddings.</figcaption></figure>

• [2], [2], [2]: three simple cusps will be shown to have a unique symplectic embedding into $\mathbb{CP}^2$ in Proposition 7.2.

As it turns out, all these configurations are realized by rational cuspidal quartics in $\mathbb{CP}^2$, and each realization has a unique equisingular isotopy class in $\mathbb{CP}^2$. The bicuspidal quartic also has realizations in $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ and $S^2 \times S^2$.

**PROPOSITION 7.1.** Let $C$ be a curve with normal Euler number 16 with one cusp of type (2,5) and one of type (2,3). Then the only relatively minimal symplectic embeddings of $C$ are into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, and $S^2 \times S^2$. For each of these there is a unique non-empty symplectic isotopy class of embeddings up to symplectomorphism. Correspondingly, the associated contact structure $\xi_C$ has three fillings; one is a rational homology ball, and the other two have second Betti number 1 and are distinguished by their intersection forms.

*Proof.* Suppose $C$ embeds minimally symplectically in $(X, \omega)$. Blow up at the (2,3)-cusp of $C$ to the normal crossing resolution, and blow up at the (2,5)-cusp once more than its minimal resolution (not quite normal crossing). See Figure 22. The proper transform of $C$ becomes a smooth symplectic +1-sphere, so we apply McDuff’s theorem to identify it with a line in some blow up of $\mathbb{CP}^2$.

We determine the homology classes of the remaining divisors in the total transform of $C$ relative to this identification. Using Lemma 3.7, and the intersection relations, the homology classes of the divisors have three possibilities differing from each other only in the $(-2)$- and $(-3)$-spheres in the normal crossing resolution of the (2,3)-cusp. The divisors in the resolution of the (2,5)-cusp must be $h - e_1 - e_2$, $h - e_3 - e_4 - e_5$, and $e_5 - e_6$, and the $(-1)$-curve in the resolution of the (2,3)-cusp must be in the class $h - e_1 - e_3$ (up to relabeling the $e_i$). The remaining $(-2)$-sphere class must be either $e_1 - e_2$ or $e_3 - e_4$. If it is $e_1 - e_2$ there are two possibilities for the $(-3)$-sphere class: $e_3 - e_5 - e_6$ and $e_3 - e_4 - e_7$. If the $(-2)$-class is $e_3 - e_4$, the $(-3)$-class must be $e_1 - e_2 - e_7$. See Figure 22.

Observe that two of these three possibilities require seven exceptional classes and the third requires six exceptional classes. As the resolution was obtained from the cuspidal curve by blowing up six times, the two embeddings of the resolution using seven exceptional classes correspond to potential embeddings of the cuspidal curve into either $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ or $S^2 \times S^2$.

In option (B), where the $(-3)$-sphere represents $e_3 - e_4 - e_7$ and the $(-2)$-sphere represents $e_1 - e_2$, the homology of the complement of the exceptional divisors is generated by $2h - e_1 - e_2 - e_3 - e_4$ and $4h - 2e_1 - 2e_2 - 2e_3 - e_4 - e_5 - e_6 - e_7$. The intersection form generated by these two classes is

$$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$
<figure><img src="image_29.png" /><figcaption>FIGURE 22. A resolution of a rational cuspidal quartic with one cusp of type (2,3) and one of type (2,5) with the possible homological embeddings.</figcaption></figure>

• [2], [2], [2]: three simple cusps will be shown to have a unique symplectic embedding into $\mathbb{CP}^2$ in Proposition 7.2.

As it turns out, all these configurations are realized by rational cuspidal quartics in $\mathbb{CP}^2$, and each realization has a unique equisingular isotopy class in $\mathbb{CP}^2$. The bicuspidal quartic also has realizations in $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ and $S^2 \times S^2$.

**PROPOSITION 7.1.** Let $C$ be a curve with normal Euler number 16 with one cusp of type (2,5) and one of type (2,3). Then the only relatively minimal symplectic embeddings of $C$ are into $\mathbb{CP}^2$, $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, and $S^2 \times S^2$. For each of these there is a unique non-empty symplectic isotopy class of embeddings up to symplectomorphism. Correspondingly, the associated contact structure $\xi_C$ has three fillings; one is a rational homology ball, and the other two have second Betti number 1 and are distinguished by their intersection forms.

*Proof.* Suppose $C$ embeds minimally symplectically in $(X, \omega)$. Blow up at the (2,3)-cusp of $C$ to the normal crossing resolution, and blow up at the (2,5)-cusp once more than its minimal resolution (not quite normal crossing). See Figure 22. The proper transform of $C$ becomes a smooth symplectic +1-sphere, so we apply McDuff’s theorem to identify it with a line in some blow up of $\mathbb{CP}^2$.

We determine the homology classes of the remaining divisors in the total transform of $C$ relative to this identification. Using Lemma 3.7, and the intersection relations, the homology classes of the divisors have three possibilities differing from each other only in the $(-2)$- and $(-3)$-spheres in the normal crossing resolution of the (2,3)-cusp. The divisors in the resolution of the (2,5)-cusp must be $h - e_1 - e_2$, $h - e_3 - e_4 - e_5$, and $e_5 - e_6$, and the $(-1)$-curve in the resolution of the (2,3)-cusp must be in the class $h - e_1 - e_3$ (up to relabeling the $e_i$). The remaining $(-2)$-sphere class must be either $e_1 - e_2$ or $e_3 - e_4$. If it is $e_1 - e_2$ there are two possibilities for the $(-3)$-sphere class: $e_3 - e_5 - e_6$ and $e_3 - e_4 - e_7$. If the $(-2)$-class is $e_3 - e_4$, the $(-3)$-class must be $e_1 - e_2 - e_7$. See Figure 22.

Observe that two of these three possibilities require seven exceptional classes and the third requires six exceptional classes. As the resolution was obtained from the cuspidal curve by blowing up six times, the two embeddings of the resolution using seven exceptional classes correspond to potential embeddings of the cuspidal curve into either $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ or $S^2 \times S^2$.

In option (B), where the $(-3)$-sphere represents $e_3 - e_4 - e_7$ and the $(-2)$-sphere represents $e_1 - e_2$, the homology of the complement of the exceptional divisors is generated by $2h - e_1 - e_2 - e_3 - e_4$ and $4h - 2e_1 - 2e_2 - 2e_3 - e_4 - e_5 - e_6 - e_7$. The intersection form generated by these two classes is

$$ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $$
so this corresponds to an embedding of the bicuspidal quartic into $S^2 \times S^2$.

In option (C), where the (-3)-sphere represents $e_1 - e_2 - e_7$ and the (-2)-sphere represents $e_3 - e_4$, the homology of the complement of the exceptional divisors in the resolution is generated by $3h - 2e_1 - e_2 - e_3 - e_4 - e_5 - e_6 - e_7$ and $5h - 3e_1 - 2e_2 - 2e_3 - 2e_4 - e_5 - e_6 - e_7$ which gives an intersection form

$$ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $$

so this corresponds to an embedding of the bicuspidal quartic into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$.

The embedding of the resolution using six $e_i$ classes (option (A)) corresponds to a potential embedding of the cuspidal curve into $\mathbb{CP}^2$.

To verify that there is a unique isotopy class for each homology embedding, we apply Lemma 3.5 to find exceptional spheres in the $e_i$ classes. In all three cases, the configuration blows down to four lines with one triple point and three double points. This has a unique symplectic isotopy class by Proposition 5.1, so this cuspidal quartic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18.

The complement of the embedding of the cuspidal curve into $\mathbb{CP}^2$ is a rational homology ball filling. We can further check that the fillings complementary to the embeddings of the cuspidal curve into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ and $S^2 \times S^2$ are, in fact, differentiated by their intersection forms. As each has $b_2 = 1$, we simply need to find the self-intersection number of a primitive class orthogonal to all of the divisors in the cap (the exceptional divisors together with the proper transform which represents $h$). In option (B), the embedding into $S^2 \times S^2$, such a class is represented by $e_4 - e_5 - e_6 - e_7$ which has self-intersection number -4 (and actually is represented by a symplectic sphere which can be blown down to get the rational homology ball filling). In option (C), the embedding into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, such a class is given by $e_1 - e_2 - e_3 - e_4 + 2e_5 + 2e_6 + 2e_7$ which has self-intersection number -16. Therefore, the two fillings are not homeomorphic. □

Finally, we turn to the case of three cusps of type (2,3). We call $E_1$, $E_2$, and $E_3$ the three exceptional divisors in the minimal resolution of the curve.

**PROPOSITION 7.2.** Let $C$ be a curve with normal Euler number 16 with three cusps of type (2,3). Then the only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this is unique up to symplectic isotopy. Correspondingly, the associated contact structure $\xi_C$ has a unique minimal symplectic filling, and it is a rational homology ball.

*Proof.* Suppose $C$ embeds minimally symplectically in $(X, \omega)$. Blow up at each cusp twice giving a resolution in between the minimal resolution and the normal crossing resolution. See Figure 23. The proper transform of $C$ becomes a smooth symplectic +1-sphere, so we apply McDuff's theorem to identify it with a line in some blow up of $\mathbb{CP}^2$, and determine the homology classes of the exceptional divisors relative to this identification. The exceptional divisors from the first two cusps are uniquely determined up to relabeling the $e_i$, and there are three options for the exceptional divisors for the third cusp as shown in Figure 23.

Next, we blow down exceptional spheres in the $e_i$ classes in each case using Lemma 3.5. The two homological embeddings utilizing seven $e_i$ classes (options (A) and (B)) both blow down to a Fano configuration of seven lines with seven triple points (three are already in the resolution and four more come from blowing down $e_1, e_2, e_3, e_4$). The Fano configuration is obstructed from realization by symplectic (or even smooth) lines by Theorem 5.20. As both embeddings of
so this corresponds to an embedding of the bicuspidal quartic into $S^2 \times S^2$.

In option (C), where the (-3)-sphere represents $e_1 - e_2 - e_7$ and the (-2)-sphere represents $e_3 - e_4$, the homology of the complement of the exceptional divisors in the resolution is generated by $3h - 2e_1 - e_2 - e_3 - e_4 - e_5 - e_6 - e_7$ and $5h - 3e_1 - 2e_2 - 2e_3 - 2e_4 - e_5 - e_6 - e_7$ which gives an intersection form

$$ \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} $$

so this corresponds to an embedding of the bicuspidal quartic into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$.

The embedding of the resolution using six $e_i$ classes (option (A)) corresponds to a potential embedding of the cuspidal curve into $\mathbb{CP}^2$.

To verify that there is a unique isotopy class for each homology embedding, we apply Lemma 3.5 to find exceptional spheres in the $e_i$ classes. In all three cases, the configuration blows down to four lines with one triple point and three double points. This has a unique symplectic isotopy class by Proposition 5.1, so this cuspidal quartic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18.

The complement of the embedding of the cuspidal curve into $\mathbb{CP}^2$ is a rational homology ball filling. We can further check that the fillings complementary to the embeddings of the cuspidal curve into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$ and $S^2 \times S^2$ are, in fact, differentiated by their intersection forms. As each has $b_2 = 1$, we simply need to find the self-intersection number of a primitive class orthogonal to all of the divisors in the cap (the exceptional divisors together with the proper transform which represents $h$). In option (B), the embedding into $S^2 \times S^2$, such a class is represented by $e_4 - e_5 - e_6 - e_7$ which has self-intersection number $-4$ (and actually is represented by a symplectic sphere which can be blown down to get the rational homology ball filling). In option (C), the embedding into $\mathbb{CP}^2 \# \overline{\mathbb{CP}}^2$, such a class is given by $e_1 - e_2 - e_3 - e_4 + 2e_5 + 2e_6 + 2e_7$ which has self-intersection number $-16$. Therefore, the two fillings are not homeomorphic. □

Finally, we turn to the case of three cusps of type (2,3). We call $E_1$, $E_2$, and $E_3$ the three exceptional divisors in the minimal resolution of the curve.

**PROPOSITION 7.2.** Let $C$ be a curve with normal Euler number 16 with three cusps of type (2,3). Then the only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this is unique up to symplectic isotopy. Correspondingly, the associated contact structure $\xi_C$ has a unique minimal symplectic filling, and it is a rational homology ball.

*Proof.* Suppose $C$ embeds minimally symplectically in $(X, \omega)$. Blow up at each cusp twice giving a resolution in between the minimal resolution and the normal crossing resolution. See Figure 23. The proper transform of $C$ becomes a smooth symplectic +1-sphere, so we apply McDuff's theorem to identify it with a line in some blow up of $\mathbb{CP}^2$, and determine the homology classes of the exceptional divisors relative to this identification. The exceptional divisors from the first two cusps are uniquely determined up to relabeling the $e_i$, and there are three options for the exceptional divisors for the third cusp as shown in Figure 23.

Next, we blow down exceptional spheres in the $e_i$ classes in each case using Lemma 3.5. The two homological embeddings utilizing seven $e_i$ classes (options (A) and (B)) both blow down to a Fano configuration of seven lines with seven triple points (three are already in the resolution and four more come from blowing down $e_1, e_2, e_3, e_4$). The Fano configuration is obstructed from realization by symplectic (or even smooth) lines by Theorem 5.20. As both embeddings of
<figure><img src="image_30.png" /><figcaption>FIGURE 23. A resolution of a rational cuspidal quartic with three cusps of type (2,3) with the possible homological embeddings.</figcaption></figure>

the cuspidal curve into any symplectic manifold with $b_2 = 2$ have birational derivations to the Fano configuration, they cannot exist by Proposition 3.17.

The last homological embedding (option (C)) uses six $e_i$ classes so because the resolution was obtained from the cuspidal curve by performing six blow-ups, this option corresponds to an embedding of the cuspidal curve into $\mathbb{CP}^2$. We now verify this exists uniquely, by blowing down exceptional spheres in $e_i$ classes using Lemma 3.5. In this case, the configuration blows down to seven lines intersecting in 6 triple points and two double points. This can be built up line by line using Proposition 5.1. Therefore, it has a unique symplectic isotopy class so there is a unique symplectic isotopy class of $C$ in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

## 7.2 Quintics

A quintic $C$ in $\mathbb{CP}^2$ has arithmetic genus $p_a(C) = \frac{1}{2}(5-1)(5-2) = 6$. This allows for a large number of potential ways to absorb that genus into cusps. Some of these collections of singularity types are realized by rational cuspidal complex curves of degree five, and others cannot be realized. Here we prove that each equisingular type which is realized by complex curves, has a unique symplectic isotopy class of realizations, and that equisingular types which do not admit a complex realization do not admit a symplectic realization either.

To prove Theorem 1.3 for quintics, and the classifications of embeddings into other closed symplectic manifolds, we first check what are the possible multiplicity multi-sequences which yield a rational cuspidal curve of arithmetic genus $p_a(C) = 6$.

If $C$ is rational and cuspidal, then $p_a(C) = \sum \delta(p)$, where the sum is taken over all singular points of $C$. By (2.3), summing $(1/2)m(m-1)$ over all multiplicity sequences of singular points, we have to obtain six; this leaves the following possibilities to arise as multiplicity multi-sequences of the singularities of $C$: [[4]], [[3, 3]], [[3, 2, 2, 2]], and [[2, 2, 2, 2, 2, 2]].

As observed in §2.1, not all sequences of integers correspond to multiplicity sequences of singularities. In particular, among the possible sequences coming from the list above, [3, 2, 2] and [3, 2, 2, 2] are not allowed. This leaves the following possibilities for each individual singularity:

* [4], which corresponds to the singularity of type (4, 5);

* $[3^k]$, which corresponds to the singularity of type $(3, 3k+1)$ ($k=1, 2$);

* $[3, 2]$, which corresponds to the singularity of type (3, 5);

* $[2^k]$, which corresponds to the singularity of type $(2, 2k+1)$, ($k=1, \dots, 6$).

In particular, all singularities we encounter here have one Puiseux pair. Starting from degree six on, there will appear curves with multiple Puiseux pairs. For instance, in degree six, there is a unicuspidal curve whose singularity has multiplicity sequence $[4, 2, 2, 2, 2]$ (this is curve $C_2$).
<figure><img src="image_30.png" /><figcaption>FIGURE 23. A resolution of a rational cuspidal quartic with three cusps of type (2,3) with the possible homological embeddings.</figcaption></figure>

the cuspidal curve into any symplectic manifold with $b_2 = 2$ have birational derivations to the Fano configuration, they cannot exist by Proposition 3.17.

The last homological embedding (option (C)) uses six $e_i$ classes so because the resolution was obtained from the cuspidal curve by performing six blow-ups, this option corresponds to an embedding of the cuspidal curve into $\mathbb{CP}^2$. We now verify this exists uniquely, by blowing down exceptional spheres in $e_i$ classes using Lemma 3.5. In this case, the configuration blows down to seven lines intersecting in 6 triple points and two double points. This can be built up line by line using Proposition 5.1. Therefore, it has a unique symplectic isotopy class so there is a unique symplectic isotopy class of $C$ in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

## 7.2 Quintics

A quintic $C$ in $\mathbb{CP}^2$ has arithmetic genus $p_a(C) = \frac{1}{2}(5-1)(5-2) = 6$. This allows for a large number of potential ways to absorb that genus into cusps. Some of these collections of singularity types are realized by rational cuspidal complex curves of degree five, and others cannot be realized. Here we prove that each equisingular type which is realized by complex curves, has a unique symplectic isotopy class of realizations, and that equisingular types which do not admit a complex realization do not admit a symplectic realization either.

To prove Theorem 1.3 for quintics, and the classifications of embeddings into other closed symplectic manifolds, we first check what are the possible multiplicity multi-sequences which yield a rational cuspidal curve of arithmetic genus $p_a(C) = 6$.

If $C$ is rational and cuspidal, then $p_a(C) = \sum \delta(p)$, where the sum is taken over all singular points of $C$. By (2.3), summing $(1/2)m(m-1)$ over all multiplicity sequences of singular points, we have to obtain six; this leaves the following possibilities to arise as multiplicity multi-sequences of the singularities of $C$: [[4]], [[3, 3]], [[3, 2, 2, 2]], and [[2, 2, 2, 2, 2, 2]].

As observed in §2.1, not all sequences of integers correspond to multiplicity sequences of singularities. In particular, among the possible sequences coming from the list above, [3, 2, 2] and [3, 2, 2, 2] are not allowed. This leaves the following possibilities for each individual singularity:

* [4], which corresponds to the singularity of type (4, 5);

* $[3^k]$, which corresponds to the singularity of type $(3, 3k+1)$ ($k=1, 2$);

* $[3, 2]$, which corresponds to the singularity of type (3, 5);

* $[2^k]$, which corresponds to the singularity of type $(2, 2k+1)$, ($k=1, \dots, 6$).

In particular, all singularities we encounter here have one Puiseux pair. Starting from degree six on, there will appear curves with multiple Puiseux pairs. For instance, in degree six, there is a unicuspidal curve whose singularity has multiplicity sequence $[4, 2, 2, 2, 2]$ (this is curve $C_2$).
M. GOLLA AND L. STARKSTON

<table><caption>TABLE 1. The first two columns contain the collections of singularities of each quintic, expressed as multiplicity sequences (MS) and as the type of their links (L). The third and fifth columns state whether they are realized as quintics in $CP^2$ or in any other rational surface, respectively. The fourth column contains the reference to [Nam84, Theorem 2.3.10] for the algebraic realization. Finally, the last column gives a reference to the relevant statement.</caption><thead><tr><th>Sing. (MS)</th><th>Sing. (L)</th><th>In CP<sup>2</sup></th><th>Namba</th><th>Fillable</th><th>Reference</th></tr></thead><tbody><tr><td>[4]</td><td>(4, 5)</td><td>Yes</td><td>1</td><td>No</td><td>Theorem 6.6</td></tr><tr><td>[3, 3]</td><td>(3, 7)</td><td>No</td><td></td><td>No</td><td>Proposition 7.3</td></tr><tr><td>[3], [3]</td><td>(3, 4), (3, 4)</td><td>No</td><td></td><td>No</td><td>Proposition 7.3</td></tr><tr><td>[3, 2], [2, 2]</td><td>(3, 5), (2, 5)</td><td>Yes</td><td>2</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3, 2], [2]<sup>2</sup></td><td>(3, 5), (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.5</td></tr><tr><td>[3], [2, 2, 2]</td><td>(3, 4), (2, 7)</td><td>Yes</td><td>4</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3], [2, 2], [2]</td><td>(3, 4), (2, 5), (2, 3)</td><td>Yes</td><td>3</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3], [2]<sup>3</sup></td><td>(3, 4), (2, 3)<sup>3</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.5</td></tr><tr><td>[2]<sup>6</sup></td><td>(2, 13)</td><td>Yes</td><td>5</td><td>Yes</td><td>Theorem 6.6</td></tr><tr><td>[2<sup>5</sup>], [2]</td><td>(2, 11), (2, 3)</td><td>No</td><td></td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>4</sup>], [2, 2]</td><td>(2, 9), (2, 5)</td><td>Yes</td><td>7</td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>4</sup>], [2]<sup>2</sup></td><td>(2, 9), (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2<sup>3</sup>], [2<sup>3</sup>]</td><td>(2, 7), (2, 7)</td><td>No</td><td></td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>3</sup>], [2, 2], [2]</td><td>(2, 7), (2, 5), (2, 3)</td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2<sup>3</sup>], [2]<sup>3</sup></td><td>(2, 7), (2, 3)<sup>3</sup></td><td>Yes</td><td>6</td><td>No</td><td>Proposition 7.12</td></tr><tr><td>[2, 2]<sup>3</sup></td><td>(2, 5)<sup>3</sup></td><td>Yes</td><td>8</td><td>No</td><td>Proposition 7.11</td></tr><tr><td>[2, 2]<sup>2</sup>, [2]<sup>2</sup></td><td>(2, 5)<sup>2</sup>, (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2, 2], [2]<sup>4</sup></td><td>(2, 5), (2, 3)<sup>4</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.7</td></tr><tr><td>[2]<sup>6</sup></td><td>(2, 3)<sup>6</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.7</td></tr></tbody></table>

in [Fen99, Corollary 1.5]); the link of this singularity is the iterated torus knot $T(2, 3; 2, 17)$ (i.e. the $(2, 17)$-cable of $T(2, 3)$).

Theorem 1.3 for degree five results from analyzing all the possibilities, which we do in the propositions making up the rest of this subsection. We provide a table summarizing the results and giving references (Table 1).

For the multi-sequence [[3, 3]], as mentioned in Example 2.18, the semigroup condition obstructs such quintic curves from being realized even smoothly in $CP^2$. For the two possible cusp types with this multi-sequence, we prove with other techniques that there is no symplectic embedding of these curves in any closed symplectic manifold. Our proof actually shows that a resolution of such a curve admits no homological embedding whose components satisfy the adjunction formula into a blow-up of $CP^2$ when we identify a +1-sphere in the resolution with the line. This is the crudest type of obstruction from the perspective of this paper, occurring already at the level of homology.

**PROPOSITION 7.3.** There is no symplectic embedding into any closed symplectic manifold of a rational cuspidal curve $C$ with normal Euler number 25 and with a single cusp of type (3, 7) or with two cusps of type (3, 4). Thus, the corresponding contact manifolds have no symplectic fillings. In particular, there is no symplectic rational cuspidal quintic in $CP^2$ with one singularity of type (3, 7) or two of type (3, 4).

*Proof.* For the case of a single cusp of type (3, 7), we blow up the normal crossing resolution three additional times, and obtain the configuration shown in Figure 24.
M. GOLLA AND L. STARKSTON

<table><caption>TABLE 1. The first two columns contain the collections of singularities of each quintic, expressed as multiplicity sequences (MS) and as the type of their links (L). The third and fifth columns state whether they are realized as quintics in $CP^2$ or in any other rational surface, respectively. The fourth column contains the reference to [Nam84, Theorem 2.3.10] for the algebraic realization. Finally, the last column gives a reference to the relevant statement.</caption><thead><tr><th>Sing. (MS)</th><th>Sing. (L)</th><th>In CP<sup>2</sup></th><th>Namba</th><th>Fillable</th><th>Reference</th></tr></thead><tbody><tr><td>[4]</td><td>(4, 5)</td><td>Yes</td><td>1</td><td>No</td><td>Theorem 6.6</td></tr><tr><td>[3, 3]</td><td>(3, 7)</td><td>No</td><td></td><td>No</td><td>Proposition 7.3</td></tr><tr><td>[3], [3]</td><td>(3, 4), (3, 4)</td><td>No</td><td></td><td>No</td><td>Proposition 7.3</td></tr><tr><td>[3, 2], [2, 2]</td><td>(3, 5), (2, 5)</td><td>Yes</td><td>2</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3, 2], [2]<sup>2</sup></td><td>(3, 5), (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.5</td></tr><tr><td>[3], [2, 2, 2]</td><td>(3, 4), (2, 7)</td><td>Yes</td><td>4</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3], [2, 2], [2]</td><td>(3, 4), (2, 5), (2, 3)</td><td>Yes</td><td>3</td><td>No</td><td>Proposition 7.4</td></tr><tr><td>[3], [2]<sup>3</sup></td><td>(3, 4), (2, 3)<sup>3</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.5</td></tr><tr><td>[2]<sup>6</sup></td><td>(2, 13)</td><td>Yes</td><td>5</td><td>Yes</td><td>Theorem 6.6</td></tr><tr><td>[2<sup>5</sup>], [2]</td><td>(2, 11), (2, 3)</td><td>No</td><td></td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>4</sup>], [2, 2]</td><td>(2, 9), (2, 5)</td><td>Yes</td><td>7</td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>4</sup>], [2]<sup>2</sup></td><td>(2, 9), (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2<sup>3</sup>], [2<sup>3</sup>]</td><td>(2, 7), (2, 7)</td><td>No</td><td></td><td>Yes</td><td>Proposition 7.9</td></tr><tr><td>[2<sup>3</sup>], [2, 2], [2]</td><td>(2, 7), (2, 5), (2, 3)</td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2<sup>3</sup>], [2]<sup>3</sup></td><td>(2, 7), (2, 3)<sup>3</sup></td><td>Yes</td><td>6</td><td>No</td><td>Proposition 7.12</td></tr><tr><td>[2, 2]<sup>3</sup></td><td>(2, 5)<sup>3</sup></td><td>Yes</td><td>8</td><td>No</td><td>Proposition 7.11</td></tr><tr><td>[2, 2]<sup>2</sup>, [2]<sup>2</sup></td><td>(2, 5)<sup>2</sup>, (2, 3)<sup>2</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.8</td></tr><tr><td>[2, 2], [2]<sup>4</sup></td><td>(2, 5), (2, 3)<sup>4</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.7</td></tr><tr><td>[2]<sup>6</sup></td><td>(2, 3)<sup>6</sup></td><td>No</td><td></td><td>No</td><td>Proposition 7.7</td></tr></tbody></table>

in [Fen99, Corollary 1.5]); the link of this singularity is the iterated torus knot $T(2, 3; 2, 17)$ (i.e. the $(2, 17)$-cable of $T(2, 3)$).

Theorem 1.3 for degree five results from analyzing all the possibilities, which we do in the propositions making up the rest of this subsection. We provide a table summarizing the results and giving references (Table 1).

For the multi-sequence [[3, 3]], as mentioned in Example 2.18, the semigroup condition obstructs such quintic curves from being realized even smoothly in $CP^2$. For the two possible cusp types with this multi-sequence, we prove with other techniques that there is no symplectic embedding of these curves in any closed symplectic manifold. Our proof actually shows that a resolution of such a curve admits no homological embedding whose components satisfy the adjunction formula into a blow-up of $CP^2$ when we identify a +1-sphere in the resolution with the line. This is the crudest type of obstruction from the perspective of this paper, occurring already at the level of homology.

**PROPOSITION 7.3.** There is no symplectic embedding into any closed symplectic manifold of a rational cuspidal curve $C$ with normal Euler number 25 and with a single cusp of type (3, 7) or with two cusps of type (3, 4). Thus, the corresponding contact manifolds have no symplectic fillings. In particular, there is no symplectic rational cuspidal quintic in $CP^2$ with one singularity of type (3, 7) or two of type (3, 4).

*Proof.* For the case of a single cusp of type (3, 7), we blow up the normal crossing resolution three additional times, and obtain the configuration shown in Figure 24.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

<figure><img src="image_31.png" /><figcaption>FIGURE 24. A resolution of a rational cuspidal quintic with one cusp of type (3, 7) showing that it admits no homological embedding.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>FIGURE 25. A resolution of a rational cuspidal quintic with two cusps of type (3, 4) showing that it admits no homological embedding.</figcaption></figure>

We identify the +1-curve with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ by Theorem 3.6. The homology classes of the upper chain are uniquely determined by Lemmas 3.7 and 3.12 to be

$$ (h, h - e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6). $$

The (-4)-sphere must have the form $e_{i_0} - e_{i_1} - e_{i_2} - e_{i_3}$. By Lemma 3.9 the classes $e_1, \dots, e_5$ cannot appear with positive coefficient twice. Using this and the intersections with the other classes in the chain, the (-4)-sphere represents $e_0 - e_3 - e_2 - e_1$. The remaining (-2)-sphere, $C$ must have the form $[C] = e_i - e_0$ for some $i > 3$, since $[C] \cdot (e_0 - e_1 - e_2 - e_3) = 1$ and $e_1, e_2, e_3$ cannot appear with positive coefficient again. Because $[C] \cdot (h - e_0 - e_1) = 0$, we must have $[C] = e_1 - e_0$, but then $[C] \cdot (e_1 - e_2) = -1$ when it should be zero. Therefore there is no adjunctive embedding of this resolution configuration.

For the curve with two cusps of type (3, 4), the normal crossing resolution graph is given by Figure 25.

Again, identify the +1-sphere with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. By Lemmas 3.11 and 3.12, the chain emanating to the right from the +1-sphere initially has two possibilities

$$ (h - e_0 - e_1, e_1 - e_2, e_2 - e_3) \text{ or } (h - e_0 - e_1, e_1 - e_2, e_0 - e_1). $$

The other (-1)-sphere must have the form $h - e_0 - e_4$ or $h - e_1 - e_4$ based on its intersection with the +1 and other (-1)-spheres. This rules out the second possibility for the chain emanating to the right because $e_0 - e_1$ intersects non-trivially with both options. The chain emanating to the left from the +1-sphere is uniquely determined similarly as shown in Figure 25.

However, then the two (-4)-spheres must each have one exceptional class with coefficient +1, and that class must appear with coefficient -1 in the (-1)-sphere they are adjacent to. As $e_1$ and $e_4$ have already appeared with positive coefficient, and no exceptional class can appear with positive coefficient twice by Lemma 3.9, there is no possible homological embedding. $\square$

Now we consider three related cases of rational cuspidal quintics which symplectically embed uniquely into $\mathbb{CP}^2$. Each of these curves is known to have a complex algebraic realization in $\mathbb{CP}^2$ (cases 2, 4, and 3 in [Nam84, Theorem 2.3.10], respectively).
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

<figure><img src="image_31.png" /><figcaption>FIGURE 24. A resolution of a rational cuspidal quintic with one cusp of type (3, 7) showing that it admits no homological embedding.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>FIGURE 25. A resolution of a rational cuspidal quintic with two cusps of type (3, 4) showing that it admits no homological embedding.</figcaption></figure>

We identify the +1-curve with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$ by Theorem 3.6. The homology classes of the upper chain are uniquely determined by Lemmas 3.7 and 3.12 to be

$$ (h, h - e_0 - e_1, e_1 - e_2, e_2 - e_3, e_3 - e_4, e_4 - e_5, e_5 - e_6). $$

The (-4)-sphere must have the form $e_{i_0} - e_{i_1} - e_{i_2} - e_{i_3}$. By Lemma 3.9 the classes $e_1, \dots, e_5$ cannot appear with positive coefficient twice. Using this and the intersections with the other classes in the chain, the (-4)-sphere represents $e_0 - e_3 - e_2 - e_1$. The remaining (-2)-sphere, $C$ must have the form $[C] = e_i - e_0$ for some $i > 3$, since $[C] \cdot (e_0 - e_1 - e_2 - e_3) = 1$ and $e_1, e_2, e_3$ cannot appear with positive coefficient again. Because $[C] \cdot (h - e_0 - e_1) = 0$, we must have $[C] = e_1 - e_0$, but then $[C] \cdot (e_1 - e_2) = -1$ when it should be zero. Therefore there is no adjunctive embedding of this resolution configuration.

For the curve with two cusps of type (3, 4), the normal crossing resolution graph is given by Figure 25.

Again, identify the +1-sphere with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$. By Lemmas 3.11 and 3.12, the chain emanating to the right from the +1-sphere initially has two possibilities

$$ (h - e_0 - e_1, e_1 - e_2, e_2 - e_3) \text{ or } (h - e_0 - e_1, e_1 - e_2, e_0 - e_1). $$

The other (-1)-sphere must have the form $h - e_0 - e_4$ or $h - e_1 - e_4$ based on its intersection with the +1 and other (-1)-spheres. This rules out the second possibility for the chain emanating to the right because $e_0 - e_1$ intersects non-trivially with both options. The chain emanating to the left from the +1-sphere is uniquely determined similarly as shown in Figure 25.

However, then the two (-4)-spheres must each have one exceptional class with coefficient +1, and that class must appear with coefficient -1 in the (-1)-sphere they are adjacent to. As $e_1$ and $e_4$ have already appeared with positive coefficient, and no exceptional class can appear with positive coefficient twice by Lemma 3.9, there is no possible homological embedding. $\square$

Now we consider three related cases of rational cuspidal quintics which symplectically embed uniquely into $\mathbb{CP}^2$. Each of these curves is known to have a complex algebraic realization in $\mathbb{CP}^2$ (cases 2, 4, and 3 in [Nam84, Theorem 2.3.10], respectively).
<figure><img src="image_33.png" /><figcaption>FIGURE 26. A resolution of a rational cuspidal quintic with one cusp of type (3,5) and one of type (2,5) with the only possible homological embedding.</figcaption></figure>

PROPOSITION 7.4. If a rational cuspidal curve $C$ has normal Euler number 25 and one of the following three cusp collections:

(i) one of type (3, 5) and one of type (2, 5);

(ii) one of type (3, 4) and one of type (2, 7);

(iii) one of type (3, 4), one of type (2, 5), and one of type (2, 3);

then the only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.

*Proof*. For each of three cusp types, we blow up to a resolution where the proper transform of the cuspidal curve is smooth and has self-intersection +1. We apply McDuff's theorem to identify the +1-sphere with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$.

For the curve with a (3,5)- and (2,5)-cusp take the normal crossing resolution at the (3,5)-cusp, and the minimal smooth resolution at the (2,5)-singularity, blown up one additional time as in Figure 26. Note that this resolution was obtained from the cuspidal curve by blowing up a total of seven times (there are seven exceptional divisors). Relative the identification of the +1-sphere with a line, the homology classes of the other curves are uniquely determined up to relabeling the $e_i$ by Lemma 3.7 and the intersection relations as shown in Figure 26. Note that this homological embedding uses seven $e_i$ classes, so if this resolution is minimally embedded, the ambient symplectic 4-manifold must be $\mathbb{CP}^2 \# 7\overline{\mathbb{CP}}^2$. Thus, if we return to the cuspidal curve by blowing down seven times to reverse the resolution, the cuspidal curve can only minimally symplectically embed in $\mathbb{CP}^2$. Blowing down spheres in the $e_i$ classes using Lemma 3.5 gives a birational derivation from the cuspidal curve to a line arrangement of four lines with a single triple point. This line arrangement has a unique symplectic isotopy class by Proposition 5.1. Therefore, this cuspidal quintic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18.

For the cuspidal curve with a (3,4)- and (2,7)-cusp, blow up two times more than the minimal resolution at the (3,4)-cusp, and blow up one time more than the minimal resolution at the (2,7)-cusp as in Figure 27. The homology classes relative the +1-line are uniquely determined as shown, and use seven $e_i$ classes (the same as the number of exceptional divisors in the resolution). Therefore, the cuspidal curve embeds symplectically minimally only in $\mathbb{CP}^2$. Using Lemma 3.5 to successively blow down exceptional spheres in the $e_i$ classes to $\mathbb{CP}^2$, the configuration descends to five lines $\{L_i\}$ where $L_1, L_2, L_3$ intersect at a triple point and $L_3, L_4, L_5$ intersect at a triple point, and the other intersections are generic. This has a unique symplectic isotopy class by Proposition 5.1. As this line arrangement is birationally derived from the embedding of the cuspidal curve, Proposition 3.18 implies that the cuspidal curve has a unique symplectic isotopy class in $\mathbb{CP}^2$.
<figure><img src="image_33.png" /><figcaption>FIGURE 26. A resolution of a rational cuspidal quintic with one cusp of type (3,5) and one of type (2,5) with the only possible homological embedding.</figcaption></figure>

PROPOSITION 7.4. If a rational cuspidal curve $C$ has normal Euler number 25 and one of the following three cusp collections:

(i) one of type (3, 5) and one of type (2, 5);

(ii) one of type (3, 4) and one of type (2, 7);

(iii) one of type (3, 4), one of type (2, 5), and one of type (2, 3);

then the only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.

*Proof*. For each of three cusp types, we blow up to a resolution where the proper transform of the cuspidal curve is smooth and has self-intersection +1. We apply McDuff's theorem to identify the +1-sphere with a line in $\mathbb{CP}^2 \# N\overline{\mathbb{CP}}^2$.

For the curve with a (3,5)- and (2,5)-cusp take the normal crossing resolution at the (3,5)-cusp, and the minimal smooth resolution at the (2,5)-singularity, blown up one additional time as in Figure 26. Note that this resolution was obtained from the cuspidal curve by blowing up a total of seven times (there are seven exceptional divisors). Relative the identification of the +1-sphere with a line, the homology classes of the other curves are uniquely determined up to relabeling the $e_i$ by Lemma 3.7 and the intersection relations as shown in Figure 26. Note that this homological embedding uses seven $e_i$ classes, so if this resolution is minimally embedded, the ambient symplectic 4-manifold must be $\mathbb{CP}^2 \# 7\overline{\mathbb{CP}}^2$. Thus, if we return to the cuspidal curve by blowing down seven times to reverse the resolution, the cuspidal curve can only minimally symplectically embed in $\mathbb{CP}^2$. Blowing down spheres in the $e_i$ classes using Lemma 3.5 gives a birational derivation from the cuspidal curve to a line arrangement of four lines with a single triple point. This line arrangement has a unique symplectic isotopy class by Proposition 5.1. Therefore, this cuspidal quintic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18.

For the cuspidal curve with a (3,4)- and (2,7)-cusp, blow up two times more than the minimal resolution at the (3,4)-cusp, and blow up one time more than the minimal resolution at the (2,7)-cusp as in Figure 27. The homology classes relative the +1-line are uniquely determined as shown, and use seven $e_i$ classes (the same as the number of exceptional divisors in the resolution). Therefore, the cuspidal curve embeds symplectically minimally only in $\mathbb{CP}^2$. Using Lemma 3.5 to successively blow down exceptional spheres in the $e_i$ classes to $\mathbb{CP}^2$, the configuration descends to five lines $\{L_i\}$ where $L_1, L_2, L_3$ intersect at a triple point and $L_3, L_4, L_5$ intersect at a triple point, and the other intersections are generic. This has a unique symplectic isotopy class by Proposition 5.1. As this line arrangement is birationally derived from the embedding of the cuspidal curve, Proposition 3.18 implies that the cuspidal curve has a unique symplectic isotopy class in $\mathbb{CP}^2$.
<figure><img src="image_34.png" /><figcaption>FIGURE 27. A resolution of a rational cuspidal quintic with one cusp of type (3,4) and one of type (2,7) with the only possible homological embedding.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>FIGURE 28. A resolution of a rational cuspidal quintic with one cusp of type (3,4), one of type (2,5), and one of type (2,3) with the only possible homological embedding.</figcaption></figure>

For the curve with cusps of type (3,4), (2,5), and (2,3), we blow up to the minimal resolution and then once more at each of the (3,4)- and (2,5)-cusps resulting in Figure 28. The homology classes relative the +1-line are uniquely determined as shown. (The uniqueness requires a little additional thought in this case, because some of the curves *a priori* have other options (the (-3)-curve and the (-2)-curves disjoint from the line). However, any combinations of these other than the one shown violate the intersection relations.) The unique homological embedding uses the same number of $e_i$ classes as exceptional divisors in the resolution. Therefore, the cuspidal curve embeds symplectically minimally only in $\mathbb{CP}^2$.

We blow down exceptional spheres in the classes $e_i$, using Lemma 3.5 together with blow-downs of the images of the (-2)-spheres disjoint from the +1-sphere. (For example, after blowing down a representative of $e_4$, the image of the (-2)-sphere in the class $e_3 - e_4$ becomes an exceptional sphere representing $e_3$. The exceptional spheres representing $e_4$ and $e_3$ are sent to the same point after these two sequential blow downs.) This gives a birational derivation from the cuspidal curve to a configuration with a conic and five lines with intersections as shown in Figure 29. This configuration has a unique equisingular symplectic isotopy class by iteratively applying Proposition 5.1 starting with $Q_0$ and adding $L_1, \dots, L_5$ one at a time in order as shown in Figure 29. Therefore, this cuspidal quintic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18 because it has a birational derivation to a configuration with a unique symplectic isotopy class. □

The remaining two possible cuspidal quintics with multi-sequence [[3, 2, 2, 2]] are obstructed from being realized by complex curves in $\mathbb{CP}^2$ by the Riemann-Hurwitz formula (see Example 3.20). Because we can use $J$-holomorphic curves to recover the Riemann-Hurwitz formula for symplectic curves, the same obstruction holds in the symplectic case. However, we can actually prove the stronger statement, that these rational cuspidal curves cannot symplectically
<figure><img src="image_34.png" /><figcaption>FIGURE 27. A resolution of a rational cuspidal quintic with one cusp of type (3,4) and one of type (2,7) with the only possible homological embedding.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>FIGURE 28. A resolution of a rational cuspidal quintic with one cusp of type (3,4), one of type (2,5), and one of type (2,3) with the only possible homological embedding.</figcaption></figure>

For the curve with cusps of type (3,4), (2,5), and (2,3), we blow up to the minimal resolution and then once more at each of the (3,4)- and (2,5)-cusps resulting in Figure 28. The homology classes relative the +1-line are uniquely determined as shown. (The uniqueness requires a little additional thought in this case, because some of the curves *a priori* have other options (the (-3)-curve and the (-2)-curves disjoint from the line). However, any combinations of these other than the one shown violate the intersection relations.) The unique homological embedding uses the same number of $e_i$ classes as exceptional divisors in the resolution. Therefore, the cuspidal curve embeds symplectically minimally only in $\mathbb{CP}^2$.

We blow down exceptional spheres in the classes $e_i$, using Lemma 3.5 together with blow-downs of the images of the (-2)-spheres disjoint from the +1-sphere. (For example, after blowing down a representative of $e_4$, the image of the (-2)-sphere in the class $e_3 - e_4$ becomes an exceptional sphere representing $e_3$. The exceptional spheres representing $e_4$ and $e_3$ are sent to the same point after these two sequential blow downs.) This gives a birational derivation from the cuspidal curve to a configuration with a conic and five lines with intersections as shown in Figure 29. This configuration has a unique equisingular symplectic isotopy class by iteratively applying Proposition 5.1 starting with $Q_0$ and adding $L_1, \dots, L_5$ one at a time in order as shown in Figure 29. Therefore, this cuspidal quintic has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18 because it has a birational derivation to a configuration with a unique symplectic isotopy class. $\square$

The remaining two possible cuspidal quintics with multi-sequence [[3, 2, 2, 2]] are obstructed from being realized by complex curves in $\mathbb{CP}^2$ by the Riemann-Hurwitz formula (see Example 3.20). Because we can use $J$-holomorphic curves to recover the Riemann-Hurwitz formula for symplectic curves, the same obstruction holds in the symplectic case. However, we can actually prove the stronger statement, that these rational cuspidal curves cannot symplectically
<figure><img src="image_36.png" /></figure>

FIGURE 29. The configuration in $\mathbb{CP}^2$ resulting from blowing down $e_1, \dots, e_7$ from Figure 28. The images of the exceptional spheres are indicated by $e(i)$. This has a unique equisingular symplectic isotopy class by iteratively applying Proposition 5.1 starting with $Q_0$ and adding $L_1, \dots, L_5$ one at a time in order.

embed in any closed symplectic manifold. Moreover, reversing our argument will recover an obstruction to a symplectic Fano plane, first proven in [RS19].

**PROPOSITION 7.5.** A rational cuspidal curves $C$ with normal Euler number 25 and either:

(i) one cusp of type (3,5) and two of type (2,3); or

(ii) one cusp of type (3,4) and three of type (2,3);

has no symplectic embedding into any closed symplectic 4-manifold. Equivalently, the associated contact structure $\xi_C$ has no symplectic filling.

*Proof.* We begin with the first case where $C$ has one cusp of type (3,5) and two of type (2,3). Suppose $C$ embeds symplectically minimally into $(X, \omega)$. Blow up four times to the minimal resolution and then once more at each of the three tangency points, until $C$ has self-intersection +1, and the configuration is given as in Figure 30. Apply McDuff's theorem to identify $X\#7\overline{\mathbb{CP}}^2$ with $\overline{\mathbb{CP}}^2\#N\overline{\mathbb{CP}}^2$ sending $C$ to a line. The homology classes of the other curves are uniquely determined by Lemma 3.7 and the algebraic intersection numbers as in Figure 30. There are exactly seven $e_i$ classes appearing so if $C$ was minimally symplectically embedded in $X$, we must have $X\#7\overline{\mathbb{CP}}^2 = \overline{\mathbb{CP}}^2\#7\overline{\mathbb{CP}}^2$ so $X = \overline{\mathbb{CP}}^2$. This together with the Riemann-Hurwitz obstruction suffices to ensure $C$ does not embed into any closed symplectic manifold. However, going a bit further, if we apply Lemma 3.5 to blow down exceptional spheres representing the $e_i$, the configuration blows down to seven lines intersecting in seven triple points: the Fano plane (Figure 13). (The exceptional spheres representing $e_1, e_2, e_3, e_4$ blow down to triple point intersections between lines.) Therefore, the Fano line arrangement in $\overline{\mathbb{CP}}^2$ can be birationally derived from this cuspidal quintic.

Thus, we obtain a proof of the following corollary, independently of [RS19]. (They prove the stronger statement in the smooth, rather than symplectic, category.)

**COROLLARY 7.6.** The Fano configuration cannot be realized as a configuration of symplectic lines in $\overline{\mathbb{CP}}^2$. $\square$

The cuspidal quintic with one (3,4)-cusp and three (2,3)-cusps is obstructed similarly. This time we blow up to the minimal resolution and then blow-up one additional time at each of the three (2,3)-cusp points (a total of 7 blow-ups). The resulting configuration is shown in
<figure><img src="image_36.png" /></figure>

FIGURE 29. The configuration in $\mathbb{CP}^2$ resulting from blowing down $e_1, \dots, e_7$ from Figure 28. The images of the exceptional spheres are indicated by $e(i)$. This has a unique equisingular symplectic isotopy class by iteratively applying Proposition 5.1 starting with $Q_0$ and adding $L_1, \dots, L_5$ one at a time in order.

embed in any closed symplectic manifold. Moreover, reversing our argument will recover an
obstruction to a symplectic Fano plane, first proven in [RS19].

PROPOSITION 7.5. A rational cuspidal curves $C$ with normal Euler number 25 and either:

(i) one cusp of type (3,5) and two of type (2,3); or

(ii) one cusp of type (3,4) and three of type (2,3);

has no symplectic embedding into any closed symplectic 4-manifold. Equivalently, the associated contact structure $\xi_C$ has no symplectic filling.

*Proof*. We begin with the first case where $C$ has one cusp of type (3,5) and two of type (2,3). Suppose $C$ embeds symplectically minimally into $(X, \omega)$. Blow up four times to the minimal resolution and then once more at each of the three tangency points, until $C$ has self-intersection +1, and the configuration is given as in Figure 30. Apply McDuff's theorem to identify $X\#7\overline{\mathbb{CP}}^2$ with $\overline{\mathbb{CP}}^2\#N\overline{\mathbb{CP}}^2$ sending $C$ to a line. The homology classes of the other curves are uniquely determined by Lemma 3.7 and the algebraic intersection numbers as in Figure 30. There are exactly seven $e_i$ classes appearing so if $C$ was minimally symplectically embedded in $X$, we must have $X\#7\overline{\mathbb{CP}}^2 = \overline{\mathbb{CP}}^2\#7\overline{\mathbb{CP}}^2$ so $X = \overline{\mathbb{CP}}^2$. This together with the Riemann–Hurwitz obstruction suffices to ensure $C$ does not embed into any closed symplectic manifold. However, going a bit further, if we apply Lemma 3.5 to blow down exceptional spheres representing the $e_i$, the configuration blows down to seven lines intersecting in seven triple points: the Fano plane (Figure 13). (The exceptional spheres representing $e_1, e_2, e_3, e_4$ blow down to triple point intersections between lines.) Therefore, the Fano line arrangement in $\overline{\mathbb{CP}}^2$ can be birationally derived from this cuspidal quintic.

Thus, we obtain a proof of the following corollary, independently of [RS19]. (They prove the stronger statement in the smooth, rather than symplectic, category.)

COROLLARY 7.6. The Fano configuration cannot be realized as a configuration of symplectic lines in $\mathbb{CP}^2$. $\square$

The cuspidal quintic with one (3,4)-cusp and three (2,3)-cusps is obstructed similarly. This time we blow up to the minimal resolution and then blow-up one additional time at each of the three (2,3)-cusp points (a total of 7 blow-ups). The resulting configuration is shown in
<figure><img src="image_37.png" /><figcaption>FIGURE 30. The unique homology classes of an embedding of the blow-up of a cuspidal curve with one cusp of type (3,5) and two of type (2,3).</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>FIGURE 31. The unique homology classes of an embedding of the blow-up of a cuspidal curve with one cusp of type (3,4) and three of type (2,3).</figcaption></figure>

Figure 31. Identifying the +1-sphere with a line, the homology classes of the other curves are uniquely determined by Lemma 3.7 and the intersection relations as shown. There are exactly seven $e_i$ classes used so the only possible 4-manifold where this cuspidal curve can minimally symplectically embed is $\mathbb{CP}^2$. Blowing down using Lemma 3.5 yields a configuration consisting of the Fano configuration together with a singular cubic. As the Fano configuration cannot exist (by Theorem 5.20 or Corollary 7.6), this cuspidal quintic cannot embed into $\mathbb{CP}^2$ by Proposition 3.17. □

There are two other quintics which are obstructed from realization by complex curves in $\mathbb{CP}^2$ using Riemann-Hurwitz (Example 3.20). We can obstruct these curves from embedding symplectically into any closed symplectic 4-manifold also.

**PROPOSITION 7.7.** Let $C$ a rational cuspidal curve with normal Euler number 25, with either:

(i) one cusp of type (2,5) and four of type (2,3); or

(ii) six cusps of type (2,3).

Then $C$ does not embed symplectically into any closed symplectic 4-manifold and contact structure $\xi_C$ associated with $C$ is not symplectically fillable.
<figure><img src="image_37.png" /><figcaption>FIGURE 30. The unique homology classes of an embedding of the blow-up of a cuspidal curve with one cusp of type (3,5) and two of type (2,3).</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>FIGURE 31. The unique homology classes of an embedding of the blow-up of a cuspidal curve with one cusp of type (3,4) and three of type (2,3).</figcaption></figure>

Figure 31. Identifying the +1-sphere with a line, the homology classes of the other curves are uniquely determined by Lemma 3.7 and the intersection relations as shown. There are exactly seven $e_i$ classes used so the only possible 4-manifold where this cuspidal curve can minimally symplectically embed is $\mathbb{CP}^2$. Blowing down using Lemma 3.5 yields a configuration consisting of the Fano configuration together with a singular cubic. As the Fano configuration cannot exist (by Theorem 5.20 or Corollary 7.6), this cuspidal quintic cannot embed into $\mathbb{CP}^2$ by Proposition 3.17. □

There are two other quintics which are obstructed from realization by complex curves in $\mathbb{CP}^2$ using Riemann-Hurwitz (Example 3.20). We can obstruct these curves from embedding symplectically into any closed symplectic 4-manifold also.

**PROPOSITION 7.7.** Let $C$ a rational cuspidal curve with normal Euler number 25, with either:

(i) one cusp of type (2,5) and four of type (2,3); or

(ii) six cusps of type (2,3).

Then $C$ does not embed symplectically into any closed symplectic 4-manifold and contact structure $\xi_C$ associated with $C$ is not symplectically fillable.
<figure><img src="image_39.png" /><figcaption>FIGURE 32. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 7), one of type (2, 5), and one of type (2, 3) with the possible homological embeddings.</figcaption></figure>

*Proof.* Assume $C$ embeds in $(X, \omega)$. In both cases, we blow up six times to the minimal resolution, where the proper transform of $C$ has self-intersection $+1$. We apply McDuff's theorem and classify homological embeddings of the configurations.

Every exceptional curve which is simply tangent to the $+1$-curve will represent the homology class $2h - e_{i_1} - \cdots - e_{i_5}$ by Lemma 3.7. Any pair of such curves are disjoint, so each pair must share exactly four exceptional classes with coefficient $-1$. When there are more than three such curves, there are two ways this can happen: either all of these curves have the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$ (for all different values of $i$) or all of the conics have the form $2h - e_1 - \cdots - e_6 + e_j$ for $j \in \{1, \ldots, 6\}$ (this is possible when there are at most six such curves).

Keeping these generalities in mind, we now focus on the case with five cusps. There are five conics tangent to the $+1$-line, which must have homology classes fitting one of the two possibilities discussed previously. The remaining $(-2)$-sphere intersects one of these conics and in each case, its homology class is uniquely determined based on its intersections with the conics as in Figure 32. In the first embedding where the conics all have the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$, there are 10 exceptional classes so $X\#6\overline{CP}^2 \cong \overline{CP}^2\#10\overline{CP}^2$, so this corresponds to an embedding of the cuspidal curve in $X = \overline{CP}^2\#4\overline{CP}^2$.

Blowing down exceptional spheres in $e_i$ classes using Lemma 3.5, the configuration in $\overline{CP}^2\#10\overline{CP}^2$ descends to five conics all intersecting at four common fixed points (the images of $e_1, e_2, e_3, e_4$), with a line tangent to all five conics. This contains the configuration $G^*$ which is obstructed by Proposition 5.22. As we birationally derived an obstructed configuration, by Proposition 3.17 this homological embedding of the minimal resolution configuration into $\overline{CP}^2\#10\overline{CP}^2$ cannot be realized by a symplectic embedding (equivalently the cuspidal curve does not have a corresponding relatively minimal embedding into $\overline{CP}^2\#4\overline{CP}^2$).

In the second embedding, only six exceptional classes are used so this corresponds to a symplectic embedding of the cuspidal curve in $\overline{CP}^2$. This is obstructed by the Riemann–Hurwitz obstruction (see Example 3.20).

In the six-cuspidal case, there are six conics tangent to the $+1$-curve $C$. Again, there are two possible homological embeddings, fully determined by the two options for the classes of the conics described previously. The first case uses 10 exceptional classes and, thus, corresponds to an embedding of the cuspidal curve into $\overline{CP}^2\#4\overline{CP}^2$. Blowing down with Lemma 3.5, the configuration descends to six conics in a pencil with a line tangent to all six. This contains the configuration $G^*$ so by Propositions 5.22 and 3.17, there is no relatively minimal symplectic embedding of this cuspidal curve into $\overline{CP}^2\#4\overline{CP}^2$. The second homological embedding uses six exceptional classes and thus corresponds to an embedding of the cuspidal curve into $\overline{CP}^2$, which is obstructed by Riemann–Hurwitz (Example 3.20). $\square$

In the previous two cases, we had potential symplectic embeddings of the minimal resolution into $\overline{CP}^2\#10\overline{CP}^2$ with homology classes determined relative to the $+1$-curve being identified with
<figure><img src="image_39.png" /><figcaption>FIGURE 32. Minimal resolution of a rational cuspidal quintic with one cusp of type (2,7), one of type (2,5), and one of type (2,3) with the possible homological embeddings.</figcaption></figure>

*Proof.* Assume $C$ embeds in $(X, \omega)$. In both cases, we blow up six times to the minimal resolution, where the proper transform of $C$ has self-intersection $+1$. We apply McDuff's theorem and classify homological embeddings of the configurations.

Every exceptional curve which is simply tangent to the +1-curve will represent the homology class $2h - e_{i_1} - \cdots - e_{i_5}$ by Lemma 3.7. Any pair of such curves are disjoint, so each pair must share exactly four exceptional classes with coefficient $-1$. When there are more than three such curves, there are two ways this can happen: either all of these curves have the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$ (for all different values of $i$) or all of the conics have the form $2h - e_1 - \cdots - e_6 + e_j$ for $j \in \{1, \ldots, 6\}$ (this is possible when there are at most six such curves).

Keeping these generalities in mind, we now focus on the case with five cusps. There are five conics tangent to the +1-line, which must have homology classes fitting one of the two possibilities discussed previously. The remaining $(-2)$-sphere intersects one of these conics and in each case, its homology class is uniquely determined based on its intersections with the conics as in Figure 32. In the first embedding where the conics all have the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$, there are 10 exceptional classes so $X\#6\overline{CP}^2 \cong \overline{CP}^2\#10\overline{CP}^2$, so this corresponds to an embedding of the cuspidal curve in $X = \overline{CP}^2\#4\overline{CP}^2$.

Blowing down exceptional spheres in $e_i$ classes using Lemma 3.5, the configuration in $\overline{CP}^2\#10\overline{CP}^2$ descends to five conics all intersecting at four common fixed points (the images of $e_1, e_2, e_3, e_4$), with a line tangent to all five conics. This contains the configuration $G^*$ which is obstructed by Proposition 5.22. As we birationally derived an obstructed configuration, by Proposition 3.17 this homological embedding of the minimal resolution configuration into $\overline{CP}^2\#10\overline{CP}^2$ cannot be realized by a symplectic embedding (equivalently the cuspidal curve does not have a corresponding relatively minimal embedding into $\overline{CP}^2\#4\overline{CP}^2$).

In the second embedding, only six exceptional classes are used so this corresponds to a symplectic embedding of the cuspidal curve in $\overline{CP}^2$. This is obstructed by the Riemann–Hurwitz obstruction (see Example 3.20).

In the six-cuspidal case, there are six conics tangent to the +1-curve $C$. Again, there are two possible homological embeddings, fully determined by the two options for the classes of the conics described previously. The first case uses 10 exceptional classes and, thus, corresponds to an embedding of the cuspidal curve into $\overline{CP}^2\#4\overline{CP}^2$. Blowing down with Lemma 3.5, the configuration descends to six conics in a pencil with a line tangent to all six. This contains the configuration $G^*$ so by Propositions 5.22 and 3.17, there is no relatively minimal symplectic embedding of this cuspidal curve into $\overline{CP}^2\#4\overline{CP}^2$. The second homological embedding uses six exceptional classes and thus corresponds to an embedding of the cuspidal curve into $\overline{CP}^2$, which is obstructed by Riemann–Hurwitz (Example 3.20). $\square$

In the previous two cases, we had potential symplectic embeddings of the minimal resolution into $\overline{CP}^2\#10\overline{CP}^2$ with homology classes determined relative to the +1-curve being identified with
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

a line. Blowing down exceptional spheres in the $e_i$ classes lead to a configuration of conics and a line that we could obstruct because it contained $G^*$. Next, we give three more cases where we can obstruct all of the possible configurations of conics and lines resulting from blowing down $e_i$ classes.

PROPOSITION 7.8. Let $C$ a rational cuspidal curve with normal Euler number 25, with:

(i) one cusp of type (2,9) and two of type (2,3);

(ii) one cusp of type (2,7), one of type (2,5), and one of type (2,3); or

(iii) two cusps of type (2,5), and two of type (2,3).

Then $C$ does not embed symplectically into any closed symplectic 4-manifold and contact structure $\xi_C$ associated with $C$ is not symplectically fillable.

*Proof.* In all three cases, the minimal resolution results from six blow-ups, and the proper transform of the cuspidal curve becomes a smooth +1-sphere $C$ which we use to apply McDuff's theorem. The six exceptional divisors are either $(-1)$-curves tangent to $C$ or $(-2)$-curves disjoint from $C$. The two choices (up to relabeling the $e_i$) of homological embedding for the tangent $(-1)$-curves each determine the homology classes of the $(-2)$-spheres. The minimal resolutions with their two possible homological embeddings are shown in Figures 33–35.

For each of the three cuspidal curves, there is one homological embedding option where all of the tangent conics represent homology class of the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$ (the upper option in the figures). If we use Lemma 3.5 to blow down exceptional spheres in the $e_i$ classes, this gives a birational derivation to a configuration that contains (or equals) $G^*$. As $G^*$ is obstructed by Proposition 5.22, there cannot be any symplectic embedding of the resolution with such homology classes in $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$.

For each of the cuspidal curves we consider the remaining homological embedding, and blow down exceptional spheres representing the $e_i$ using Lemma 3.5.

For the curve with one (2,9)-cusp and two (2,3)-cusps, the minimal resolution is shown in Figure 33. Focusing on the lower homological embedding, we blow down spheres representing classes $e_6$, $e_1$, $e_2$, and $e_3$ all to the same point. Because all four of these exceptional spheres intersect the two conics on the right (which resolved the two (2,3)-cusps), the resulting configuration in the blow-down will contain two conics which intersect at a single point with multiplicity four, one more conic, and a line tangent to all three conics. Focusing on the first two conics and the tangent line, we see this configuration contains $G_4$, obstructed by Proposition 5.23. Therefore, by Proposition 3.17, the cuspidal curve with a (2,9)-cusp and two (2,3)-cusps cannot symplectically embed into $\mathbb{CP}^2$.

For the curve with one cusp each of types (2,7), (2,5), and (2,3), the minimal resolution with the possible homological embeddings appears in Figure 34. Again focusing on the lower homological embedding, which is the only one left to rule out, we blow down exceptional spheres in $e_i$ classes using Lemma 3.5. We see that exceptional spheres in classes $e_1$, $e_2$, and $e_6$ all blow down to the same point, and exceptional spheres in classes $e_3$ and $e_4$ blow down to the same point. The two leftmost conics (coming from the last resolving exceptional divisors of the (2,7)- and (2,3)-cusps) each intersect $e_1$, $e_2$, $e_3$, and $e_4$. Therefore, after blowing down these two conics become tangent at two different points and the +1-sphere descends to a line tangent to each. Therefore, $G_{2,2}$ appears as a subconfiguration of the result of blowing down. As $G_{2,2}$ is obstructed by Proposition 5.23, the cuspidal curve cannot symplectically embed because it has a birational derivation to an obstructed configuration.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

a line. Blowing down exceptional spheres in the $e_i$ classes lead to a configuration of conics and a line that we could obstruct because it contained $G^*$. Next, we give three more cases where we can obstruct all of the possible configurations of conics and lines resulting from blowing down $e_i$ classes.

PROPOSITION 7.8. Let $C$ a rational cuspidal curve with normal Euler number 25, with:

(i) one cusp of type (2,9) and two of type (2,3);

(ii) one cusp of type (2,7), one of type (2,5), and one of type (2,3); or

(iii) two cusps of type (2,5), and two of type (2,3).

Then $C$ does not embed symplectically into any closed symplectic 4-manifold and contact structure $\xi_C$ associated with $C$ is not symplectically fillable.

*Proof.* In all three cases, the minimal resolution results from six blow-ups, and the proper transform of the cuspidal curve becomes a smooth +1-sphere $C$ which we use to apply McDuff's theorem. The six exceptional divisors are either $(-1)$-curves tangent to $C$ or $(-2)$-curves disjoint from $C$. The two choices (up to relabeling the $e_i$) of homological embedding for the tangent $(-1)$-curves each determine the homology classes of the $(-2)$-spheres. The minimal resolutions with their two possible homological embeddings are shown in Figures 33–35.

For each of the three cuspidal curves, there is one homological embedding option where all of the tangent conics represent homology class of the form $2h - e_1 - e_2 - e_3 - e_4 - e_i$ (the upper option in the figures). If we use Lemma 3.5 to blow down exceptional spheres in the $e_i$ classes, this gives a birational derivation to a configuration that contains (or equals) $G^*$. As $G^*$ is obstructed by Proposition 5.22, there cannot be any symplectic embedding of the resolution with such homology classes in $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$.

For each of the cuspidal curves we consider the remaining homological embedding, and blow down exceptional spheres representing the $e_i$ using Lemma 3.5.

For the curve with one (2,9)-cusp and two (2,3)-cusps, the minimal resolution is shown in Figure 33. Focusing on the lower homological embedding, we blow down spheres representing classes $e_6$, $e_1$, $e_2$, and $e_3$ all to the same point. Because all four of these exceptional spheres intersect the two conics on the right (which resolved the two (2,3)-cusps), the resulting configuration in the blow-down will contain two conics which intersect at a single point with multiplicity four, one more conic, and a line tangent to all three conics. Focusing on the first two conics and the tangent line, we see this configuration contains $G_4$, obstructed by Proposition 5.23. Therefore, by Proposition 3.17, the cuspidal curve with a (2,9)-cusp and two (2,3)-cusps cannot symplectically embed into $\mathbb{CP}^2$.

For the curve with one cusp each of types (2,7), (2,5), and (2,3), the minimal resolution with the possible homological embeddings appears in Figure 34. Again focusing on the lower homological embedding, which is the only one left to rule out, we blow down exceptional spheres in $e_i$ classes using Lemma 3.5. We see that exceptional spheres in classes $e_1$, $e_2$, and $e_6$ all blow down to the same point, and exceptional spheres in classes $e_3$ and $e_4$ blow down to the same point. The two leftmost conics (coming from the last resolving exceptional divisors of the (2,7)- and (2,3)-cusps) each intersect $e_1$, $e_2$, $e_3$, and $e_4$. Therefore, after blowing down these two conics become tangent at two different points and the +1-sphere descends to a line tangent to each. Therefore, $G_{2,2}$ appears as a subconfiguration of the result of blowing down. As $G_{2,2}$ is obstructed by Proposition 5.23, the cuspidal curve cannot symplectically embed because it has a birational derivation to an obstructed configuration.
<figure><img src="image_40.png" /><figcaption>FIGURE 33. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 9) and two of type (2, 3) with the possible homological embeddings.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>FIGURE 34. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 7), one of type (2, 5), and one of type (2, 3) with the possible homological embeddings.</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>FIGURE 35. Minimal resolution of a rational cuspidal quintic with two cusps of type (2, 5) and two of type (2, 3) with the possible homological embeddings.</figcaption></figure>

For the curve with two (2, 5)-cusps and two (2, 3)-cusps the minimal resolution with homological embeddings is in Figure 35. To rule out the second homological embedding, we blow down using Lemma 3.5. The pairs of exceptional spheres $e_1, e_6$ and $e_2, e_3$ each blow down to a single point. The two central conics coming from exceptional divisors where the (2, 3)-cusps were blown up both intersect all four of these exceptional spheres. Therefore, these two curves descend to two conics tangent at two points with a line tangent to both, which again is the obstructed configuration $G_{2,2}$. Thus, by Propositions 5.23 and 3.17, this cuspidal curve cannot symplectically embed. □

In the next three cases, we look at the bicuspidal curves with multi-sequence [[2, 2, 2, 2, 2, 2]]. This is the first case where we find relatively minimal symplectic embeddings into a non-trivial blow-up of $\mathbb{CP}^2$. All three cases have relatively minimal symplectic embeddings into $\mathbb{CP}^2 \#4\overline{\mathbb{CP}}^2$ and one of the cases also has a symplectic embedding into $\mathbb{CP}^2$ (case 7 in [Nam84, Theorem 2.3.10]).
<figure><img src="image_40.png" /><figcaption>FIGURE 33. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 9) and two of type (2, 3) with the possible homological embeddings.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>FIGURE 34. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 7), one of type (2, 5), and one of type (2, 3) with the possible homological embeddings.</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>FIGURE 35. Minimal resolution of a rational cuspidal quintic with two cusps of type (2, 5) and two of type (2, 3) with the possible homological embeddings.</figcaption></figure>

For the curve with two (2, 5)-cusps and two (2, 3)-cusps the minimal resolution with homological embeddings is in Figure 35. To rule out the second homological embedding, we blow down using Lemma 3.5. The pairs of exceptional spheres $e_1, e_6$ and $e_2, e_3$ each blow down to a single point. The two central conics coming from exceptional divisors where the (2, 3)-cusps were blown up both intersect all four of these exceptional spheres. Therefore, these two curves descend to two conics tangent at two points with a line tangent to both, which again is the obstructed configuration $G_{2,2}$. Thus, by Propositions 5.23 and 3.17, this cuspidal curve cannot symplectically embed. □

In the next three cases, we look at the bicuspidal curves with multi-sequence [[2, 2, 2, 2, 2, 2]]. This is the first case where we find relatively minimal symplectic embeddings into a non-trivial blow-up of $\mathbb{CP}^2$. All three cases have relatively minimal symplectic embeddings into $\mathbb{CP}^2 \#4\overline{\mathbb{CP}}^2$ and one of the cases also has a symplectic embedding into $\mathbb{CP}^2$ (case 7 in [Nam84, Theorem 2.3.10]).
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

PROPOSITION 7.9. Let $C$ be a rational cuspidal curve with normal Euler number 25.

If $C$ has one cusp of type (2,9) and one of type (2,5), then $C$ has a unique minimal symplectic embedding in $\mathbb{CP}^2$ and a unique minimal symplectic embedding into $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectic isotopy and symplectomorphism. Equivalently, the corresponding contact structure $\xi_C$ has two symplectic fillings with $b_2 = 0, 4$ respectively.

If $C$ has one cusp of type (2,11) and one of type (2,3), or two cusps of type (2,7), then $C$ has a unique minimal symplectic embedding into $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectic isotopy and symplectomorphism. Equivalently, the corresponding contact structure $\xi_C$ has a unique symplectic filling with $b_2 = 4$.

*Proof*. In all three cases, we will blow up the cuspidal curve to its minimal resolution where the proper transform $C$ is a smooth $+1$-sphere that we identify with a line using McDuff's theorem. Because there are exactly two cusps, the resolution will always contain two $(-1)$-exceptional divisors tangent to the $+1$-curve and four other $(-2)$-exceptional divisors.

We determine homology classes relative the $+1$-curve. The two tangent $(-1)$-divisors necessarily must be $2h - e_1 - e_2 - e_3 - e_4 - e_5$ and $2h - e_2 - e_3 - e_4 - e_5 - e_6$ (up to relabeling the $e_i$). We then determine the options for the four $(-2)$-curves according to the intersection relations. These are shown in Figures 36–38.

In each case, there are two possible homological embeddings of the minimal resolution. One into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ and the other into $\mathbb{CP}^2 \# 10\overline{\mathbb{CP}}^2$. If these homological embeddings can be realized symplectically, they would give symplectic embeddings of the cuspidal curve into $\mathbb{CP}^2$ and $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$, respectively.

In each of the three cases, if we assume we have a relatively minimal symplectic embedding of the minimal resolution into $\mathbb{CP}^2 \# 10\overline{\mathbb{CP}}^2$, and we blow down exceptional spheres representing the $e_i$ using Lemma 3.5, the resulting configuration in $\mathbb{CP}^2$ consists of two conics intersecting transversely at four points (the disjoint images of $e_2, e_3, e_4$, and $e_5$) with a line tangent to each. This is the configuration $G_1$ which has a unique symplectic isotopy class by Proposition 5.9. Therefore, we have a birational derivation from the cuspidal curve in $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ to a configuration with a unique nonempty symplectic isotopy class so by Proposition 3.18, each of the three types of cuspidal curves has a unique relatively minimal symplectic embedding in $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectomorphism.

For the homological embedding of the minimal resolution into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$, we consider each cuspidal curve separately.

For the case with one cusp of type (2,11) and one of type (2,3), we start with a relatively minimal embedding of the resolution into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ and blow down exceptional spheres in the $e_i$ classes using Lemma 3.5. The exceptional spheres in classes $e_2, e_3, e_4, e_5$, and $e_6$ all blow down to the same point. This produces a order-four tangency between the two conics so the configuration descends to $G_4$ which is obstructed (Proposition 5.23). Therefore, this cuspidal curve does not symplectically embed in $\mathbb{CP}^2$.

Next, we start with a symplectic embedding into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ of the minimal resolution of the curve with one cusp of type (2,9) and one of type (2,5) with the second homological embedding given in Figure 37. Blowing down exceptional spheres in the $e_i$ classes, we see that the spheres in classes $e_3, e_4, e_5$, and $e_6$ descend to the same point, and those in classes $e_1$ and $e_2$ descend to another point. The effect is that the conics become tangent at one point with multiplicity three and intersect transversally at another point. This yields the configuration $G_3$ which has a unique symplectic isotopy class by Proposition 5.5. Therefore, by Proposition 3.18, there is a unique relatively minimal symplectic embedding of this cuspidal curve into $\mathbb{CP}^2$.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

PROPOSITION 7.9. Let $C$ be a rational cuspidal curve with normal Euler number 25.

If $C$ has one cusp of type (2,9) and one of type (2,5), then $C$ has a unique minimal symplectic embedding in $\mathbb{CP}^2$ and a unique minimal symplectic embedding into $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectic isotopy and symplectomorphism. Equivalently, the corresponding contact structure $\xi_C$ has two symplectic fillings with $b_2 = 0, 4$ respectively.

If $C$ has one cusp of type (2,11) and one of type (2,3), or two cusps of type (2,7), then $C$ has a unique minimal symplectic embedding into $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectic isotopy and symplectomorphism. Equivalently, the corresponding contact structure $\xi_C$ has a unique symplectic filling with $b_2 = 4$.

*Proof.* In all three cases, we will blow up the cuspidal curve to its minimal resolution where the proper transform $C$ is a smooth $+1$-sphere that we identify with a line using McDuff's theorem. Because there are exactly two cusps, the resolution will always contain two $(-1)$-exceptional divisors tangent to the $+1$-curve and four other $(-2)$-exceptional divisors.

We determine homology classes relative the $+1$-curve. The two tangent $(-1)$-divisors necessarily must be $2h - e_1 - e_2 - e_3 - e_4 - e_5$ and $2h - e_2 - e_3 - e_4 - e_5 - e_6$ (up to relabeling the $e_i$). We then determine the options for the four $(-2)$-curves according to the intersection relations. These are shown in Figures 36–38.

In each case, there are two possible homological embeddings of the minimal resolution. One into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ and the other into $\mathbb{CP}^2 \# 10\overline{\mathbb{CP}}^2$. If these homological embeddings can be realized symplectically, they would give symplectic embeddings of the cuspidal curve into $\mathbb{CP}^2$ and $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$, respectively.

In each of the three cases, if we assume we have a relatively minimal symplectic embedding of the minimal resolution into $\mathbb{CP}^2 \# 10\overline{\mathbb{CP}}^2$, and we blow down exceptional spheres representing the $e_i$ using Lemma 3.5, the resulting configuration in $\mathbb{CP}^2$ consists of two conics intersecting transversely at four points (the disjoint images of $e_2, e_3, e_4$, and $e_5$) with a line tangent to each. This is the configuration $G_1$ which has a unique symplectic isotopy class by Proposition 5.9. Therefore, we have a birational derivation from the cuspidal curve in $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ to a configuration with a unique nonempty symplectic isotopy class so by Proposition 3.18, each of the three types of cuspidal curves has a unique relatively minimal symplectic embedding in $\mathbb{CP}^2 \# 4\overline{\mathbb{CP}}^2$ up to symplectomorphism.

For the homological embedding of the minimal resolution into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$, we consider each cuspidal curve separately.

For the case with one cusp of type (2,11) and one of type (2,3), we start with a relatively minimal embedding of the resolution into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ and blow down exceptional spheres in the $e_i$ classes using Lemma 3.5. The exceptional spheres in classes $e_2, e_3, e_4, e_5$, and $e_6$ all blow down to the same point. This produces a order-four tangency between the two conics so the configuration descends to $G_4$ which is obstructed (Proposition 5.23). Therefore, this cuspidal curve does not symplectically embed in $\mathbb{CP}^2$.

Next, we start with a symplectic embedding into $\mathbb{CP}^2 \# 6\overline{\mathbb{CP}}^2$ of the minimal resolution of the curve with one cusp of type (2,9) and one of type (2,5) with the second homological embedding given in Figure 37. Blowing down exceptional spheres in the $e_i$ classes, we see that the spheres in classes $e_3, e_4, e_5$, and $e_6$ descend to the same point, and those in classes $e_1$ and $e_2$ descend to another point. The effect is that the conics become tangent at one point with multiplicity three and intersect transversally at another point. This yields the configuration $G_3$ which has a unique symplectic isotopy class by Proposition 5.5. Therefore, by Proposition 3.18, there is a unique relatively minimal symplectic embedding of this cuspidal curve into $\mathbb{CP}^2$.
<figure><img src="image_43.png" /><figcaption>FIGURE 36. Minimal smooth resolution of a rational cuspidal quintic with one cusp of type (2, 11) and one of type (2, 3).</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>FIGURE 37. Minimal smooth resolution of a rational cuspidal quintic with one cusp of type (2, 9) and one of type (2, 5).</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>FIGURE 38. Minimal smooth resolution of a rational cuspidal quintic with two cusps of type (2, 7).</figcaption></figure>

Finally, considering a potential symplectic embedding into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$ of the minimal resolution of the cuspidal quintic with two cusps of type (2, 7), the homological embedding is given in Figure 38. Blowing down exceptional spheres using Lemma 3.5 we find that exceptional spheres in classes $e_4$, $e_5$, and $e_6$ descend to one point and those in classes $e_1$, $e_2$, and $e_3$ descend to another point. This has the effect that the two conics intersect tangentially at two points. Therefore, the configuration descends to $\mathcal{G}_{2,2}$ which is obstructed by Proposition 5.23. Thus, by Proposition 3.17, there is no symplectic embedding of this cuspidal quintic in $\mathbb{CP}^2$. $\square$

*Remark 7.10.* The rational cuspidal quintic with two cusps of type (2, 7) can be alternatively be obstructed in $\mathbb{CP}^2$ using spectrum semicontinuity as in Example 2.20.

Finally, there are two more possible collections of cusps with multi-sequence [[2, 2, 2, 2, 2, 2]]. There can be three cusps of type (2, 5) or one of type (2, 7) and three of type (2, 3). Each of these cases turns out to have a unique symplectic embedding into $\mathbb{CP}^2$ and no other relatively minimal
<figure><img src="image_43.png" /><figcaption>FIGURE 36. Minimal smooth resolution of a rational cuspidal quintic with one cusp of type (2, 11) and one of type (2, 3).</figcaption></figure>

<figure><img src="image_44.png" /><figcaption>FIGURE 37. Minimal smooth resolution of a rational cuspidal quintic with one cusp of type (2, 9) and one of type (2, 5).</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>FIGURE 38. Minimal smooth resolution of a rational cuspidal quintic with two cusps of type (2, 7).</figcaption></figure>

Finally, considering a potential symplectic embedding into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$ of the minimal resolution of the cuspidal quintic with two cusps of type (2, 7), the homological embedding is given in Figure 38. Blowing down exceptional spheres using Lemma 3.5 we find that exceptional spheres in classes $e_4$, $e_5$, and $e_6$ descend to one point and those in classes $e_1$, $e_2$, and $e_3$ descend to another point. This has the effect that the two conics intersect tangentially at two points. Therefore, the configuration descends to $\mathcal{G}_{2,2}$ which is obstructed by Proposition 5.23. Thus, by Proposition 3.17, there is no symplectic embedding of this cuspidal quintic in $\mathbb{CP}^2$. □

*Remark 7.10.* The rational cuspidal quintic with two cusps of type (2, 7) can be alternatively be obstructed in $\mathbb{CP}^2$ using spectrum semicontinuity as in Example 2.20.

Finally, there are two more possible collections of cusps with multi-sequence [[2, 2, 2, 2, 2, 2]]. There can be three cusps of type (2, 5) or one of type (2, 7) and three of type (2, 3). Each of these cases turns out to have a unique symplectic embedding into $\mathbb{CP}^2$ and no other relatively minimal
<figure><img src="image_46.png" /><figcaption>FIGURE 39. Minimal resolution of a rational cuspidal quintic with three cusps of type (2, 5) with the possible homological embeddings.</figcaption></figure>

symplectic embeddings (cases 8 and 6 in [Nam84, Theorem 2.3.10], respectively). Some of the reducible configurations of conics and lines that appear in these proofs required more extensive arguments to establish their uniqueness in § 5, so we separate these two cases as the trickiest of the bunch.

**PROPOSITION 7.11.** If a rational cuspidal curve $C$ has normal Euler number 25 and three cusps of type (2, 5), then only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.

*Proof.* We look at the minimal smooth resolution at each singularity of $C$; the proper transform of $C$ is a smooth $+1$-sphere, and the total transform is given in Figure 39 with the possible homological embeddings where the $+1$-sphere is identified with a line.

Blowing down exceptional spheres in $e_i$ classes with Lemma 3.5, the homological embedding of the resolution into $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$ blows down to the configuration $G^*$. Owing to this birational derivation, Propositions 3.17 and 5.22 imply this minimal embedding is not realizable.

The other homological embedding is of the minimal resolution into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$. After blowing down $e_i$ spheres we obtain three conics with three intersection points, such that each pair of them are tangent at a distinct intersection point, plus one line tangent to each of the three conics at a different point.

We are better equipped to work with configurations with a single conic and many lines than many conics and a single line. Therefore, we will find a birationally equivalent configuration. First add one line through each of the three pairs of intersection points of the conics (each intersecting the tangent line generically in double points). This does not change the symplectic isotopy classification of the configuration by Proposition 5.1. The resulting configuration is birationally equivalent to the configuration $\mathcal{H}$ of one conic with six lines intersecting in triple points as shown in Figure 11. The birational equivalence comes from blowing up once at each of the three intersection points of the conics and then blowing down the proper transforms of the three added lines. Here $\mathcal{H}$ has a unique equisingular symplectic isotopy class by Proposition 5.14, so this cuspidal quintic also has a unique equisingular symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

**PROPOSITION 7.12.** If a rational cuspidal curve $C$ has normal Euler number 25 and one cusp of type (2, 7) and three of type (2, 3), then only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

<figure><img src="image_46.png" /><figcaption>FIGURE 39. Minimal resolution of a rational cuspidal quintic with three cusps of type (2,5) with the possible homological embeddings.</figcaption></figure>

symplectic embeddings (cases 8 and 6 in [Nam84, Theorem 2.3.10], respectively). Some of the reducible configurations of conics and lines that appear in these proofs required more extensive arguments to establish their uniqueness in § 5, so we separate these two cases as the trickiest of the bunch.

**PROPOSITION 7.11.** If a rational cuspidal curve $C$ has normal Euler number 25 and three cusps of type (2,5), then only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.

*Proof.* We look at the minimal smooth resolution at each singularity of $C$; the proper transform of $C$ is a smooth $+1$-sphere, and the total transform is given in Figure 39 with the possible homological embeddings where the $+1$-sphere is identified with a line.

Blowing down exceptional spheres in $e_i$ classes with Lemma 3.5, the homological embedding of the resolution into $\mathbb{CP}^2\#10\overline{\mathbb{CP}}^2$ blows down to the configuration $G^*$. Owing to this birational derivation, Propositions 3.17 and 5.22 imply this minimal embedding is not realizable.

The other homological embedding is of the minimal resolution into $\mathbb{CP}^2\#6\overline{\mathbb{CP}}^2$. After blowing down $e_i$ spheres we obtain three conics with three intersection points, such that each pair of them are tangent at a distinct intersection point, plus one line tangent to each of the three conics at a different point.

We are better equipped to work with configurations with a single conic and many lines than many conics and a single line. Therefore, we will find a birationally equivalent configuration. First add one line through each of the three pairs of intersection points of the conics (each intersecting the tangent line generically in double points). This does not change the symplectic isotopy classification of the configuration by Proposition 5.1. The resulting configuration is birationally equivalent to the configuration $\mathcal{H}$ of one conic with six lines intersecting in triple points as shown in Figure 11. The birational equivalence comes from blowing up once at each of the three intersection points of the conics and then blowing down the proper transforms of the three added lines. Here $\mathcal{H}$ has a unique equisingular symplectic isotopy class by Proposition 5.14, so this cuspidal quintic also has a unique equisingular symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

**PROPOSITION 7.12.** If a rational cuspidal curve $C$ has normal Euler number 25 and one cusp of type (2,7) and three of type (2,3), then only relatively minimal symplectic embedding of $C$ is into $\mathbb{CP}^2$ and this embedding is unique up to symplectic isotopy. Equivalently, the corresponding contact structure $\xi_C$ has a unique minimal filling which is a rational homology ball.
<figure><img src="image_47.png" /><figcaption>FIGURE 40. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 7) and three of type (2, 3) with the possible homological embeddings.</figcaption></figure>

*Proof.* In the minimal resolution, the proper transform of $C$ is smooth with self-intersection $+1$ so we use Theorem 3.6 to identify it with a line and classify the homological embeddings of the exceptional divisors in Figure 40.

As previously, using Lemma 3.5 to blow down $e_i$ exceptional spheres, the first homological embedding blows down to a configuration that contains $G^*$, so it leads to no embedding by Proposition 5.22.

The second homological embedding uses exactly six exceptional classes therefore corresponds to an embedding of the cuspidal curve in $\mathbb{CP}^2$. Blowing down exceptional spheres in $e_i$ classes in the second homological embedding yields a configuration of four conics and a line. Let $p_0$ denote the image of $e_1, e_2$, and $e_3$ after the blow-down. At $p_0$, three of the conics $Q_1, Q_2, Q_3$ intersect tangentially with multiplicity three and the fourth conic $Q_4$ is tangent to these three with multiplicity two. There are three transverse triple intersections of $Q_i, Q_j, Q_4$ for the three pairs $\{i, j\} \subset \{1, 2, 3\}$. The line $L$ is tangent to all four conics, but does not go through any of the intersections of the $Q_i$.

We again look for a birational equivalence to a configuration with fewer conics and more lines. To ensure we get a birational equivalence, we first add a line $T$ tangent to the four conics at $p_0$. This does not change the symplectic isotopy uniqueness by Proposition 5.1. Next we blow up three times at $p_0$ to separate the conics at that point. The proper transform of $T$ becomes a $(-1)$-exceptional sphere which we can blow down, and then we blow down two more $(-1)$-exceptional spheres as in Figure 41. The result is the configuration $\mathcal{L}$ together with an additional line. Here $\mathcal{L}$ is made up of two conics $P_1, P_2$ with two tangencies, together with three lines such that each line is tangent to $P_1$ and the pairwise intersections of the lines are three distinct points on $P_2$. The line $K$ that is added to $\mathcal{L}$ as the result of this birational equivalence is tangent to the two conics at one of their tangent intersection points. The symplectic isotopy classifications of $\mathcal{L}$ and $\mathcal{L} \cup K$ are equivalent by Proposition 5.1. As $\mathcal{L}$ has a unique symplectic isotopy class by Proposition 5.15, this cuspidal curve has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

## 7.3 A note on sextics

In degree six, there is only one multiplicity multi-sequence that passes the semigroup condition, but such that we cannot find a cap with a sphere of positive self-intersection; this multiplicity multi-sequence is [[3, 2[⁷]]].

**PROPOSITION 7.13.** There is no rational cuspidal curve in $\mathbb{CP}^2$ whose multiplicity multi-sequence is [[3, 2[⁷]]].

Note that, in fact, this shows that the corresponding contact structures have no rational homology ball fillings (see § 4).
<figure><img src="image_47.png" /><figcaption>FIGURE 40. Minimal resolution of a rational cuspidal quintic with one cusp of type (2, 7) and three of type (2, 3) with the possible homological embeddings.</figcaption></figure>

*Proof.* In the minimal resolution, the proper transform of $C$ is smooth with self-intersection $+1$ so we use Theorem 3.6 to identify it with a line and classify the homological embeddings of the exceptional divisors in Figure 40.

As previously, using Lemma 3.5 to blow down $e_i$ exceptional spheres, the first homological embedding blows down to a configuration that contains $G^*$, so it leads to no embedding by Proposition 5.22.

The second homological embedding uses exactly six exceptional classes therefore corresponds to an embedding of the cuspidal curve in $\mathbb{CP}^2$. Blowing down exceptional spheres in $e_i$ classes in the second homological embedding yields a configuration of four conics and a line. Let $p_0$ denote the image of $e_1, e_2$, and $e_3$ after the blow-down. At $p_0$, three of the conics $Q_1, Q_2, Q_3$ intersect tangentially with multiplicity three and the fourth conic $Q_4$ is tangent to these three with multiplicity two. There are three transverse triple intersections of $Q_i, Q_j, Q_4$ for the three pairs $\{i, j\} \subset \{1, 2, 3\}$. The line $L$ is tangent to all four conics, but does not go through any of the intersections of the $Q_i$.

We again look for a birational equivalence to a configuration with fewer conics and more lines. To ensure we get a birational equivalence, we first add a line $T$ tangent to the four conics at $p_0$. This does not change the symplectic isotopy uniqueness by Proposition 5.1. Next we blow up three times at $p_0$ to separate the conics at that point. The proper transform of $T$ becomes a $(-1)$-exceptional sphere which we can blow down, and then we blow down two more $(-1)$-exceptional spheres as in Figure 41. The result is the configuration $\mathcal{L}$ together with an additional line. Here $\mathcal{L}$ is made up of two conics $P_1, P_2$ with two tangencies, together with three lines such that each line is tangent to $P_1$ and the pairwise intersections of the lines are three distinct points on $P_2$. The line $K$ that is added to $\mathcal{L}$ as the result of this birational equivalence is tangent to the two conics at one of their tangent intersection points. The symplectic isotopy classifications of $\mathcal{L}$ and $\mathcal{L} \cup K$ are equivalent by Proposition 5.1. As $\mathcal{L}$ has a unique symplectic isotopy class by Proposition 5.15, this cuspidal curve has a unique symplectic isotopy class in $\mathbb{CP}^2$ by Proposition 3.18. $\square$

## 7.3 A note on sextics

In degree six, there is only one multiplicity multi-sequence that passes the semigroup condition, but such that we cannot find a cap with a sphere of positive self-intersection; this multiplicity multi-sequence is [[3, 2[⁷]]].

**PROPOSITION 7.13.** There is no rational cuspidal curve in $\mathbb{CP}^2$ whose multiplicity multi-sequence is [[3, 2[⁷]]].

Note that, in fact, this shows that the corresponding contact structures have no rational homology ball fillings (see § 4).
<figure><img src="image_48.png" /><figcaption>FIGURE 41. Birational equivalence.</figcaption></figure>

*Proof.* The only allowed multiplicity sequences that are subsets of [[3, 2[⁷]]] are [2[⁴]], [3], and [3, 2]; these are all simple singularities of types $A_{2k}$, $E_6$, and $E_8$, respectively; their links are $T(2, 2k + 1)$, $T(3, 4)$, and $T(3, 5)$, respectively.

Suppose such a curve exists; perturb it to get a smooth curve $C'$ of genus 10, and take the double cover $\Sigma(\mathbb{CP}^2, C')$ of $\mathbb{CP}^2$ branched over $C'$; this is a $K3$ surface, and, by construction, it contains (as pairwise disjointly embedded) the double covers of $\mathbb{B}^4$ branched over the Milnor fiber of all the singularities of $C$.

All these Milnor fibers are negative definite, because the singularities are simple, and their second Betti numbers sum to 20, which contradicts the fact that $b_2^-(K3) = 19$. $\square$

This is where our approach for sextics differs from the previous cases we examined: here we are not trying to classify fillings, but rather looking specifically at embeddings in $\mathbb{CP}^2$ or, equivalently, at rational homology ball fillings. Moreover, the same argument can be used to narrow down the number of different splittings of the multiplicity multi-sequences that one needs to consider; namely, there can be no sextic whose singularities are all simple.

*Remark 7.14.* A similar argument to that of the proposition above, using five-fold covers instead of two-fold covers, can be used to obstruct the existence of the two quintics in $\mathbb{CP}^2$ with five and six singularities (which we excluded in Example 3.20; the non-fillability of the corresponding cuspidal contact structures was established in Proposition 7.7). A general result, of a more topological flavor, has been proved in [GK22, Theorem 4.7].

## 8. Differences between the complex and symplectic categories

Every complex algebraic curve in $\mathbb{CP}^2$ is a symplectic surface (potentially singular), but most symplectic surfaces are not complex algebraic. However, in many situations, a symplectic surface is symplectically isotopic to a complex curve. This is known for smooth symplectic surfaces of degree at most 17 and is conjectured for smooth symplectic surfaces in $\mathbb{CP}^2$ in general. However,
<figure><img src="image_48.png" /><figcaption>FIGURE 41. Birational equivalence.</figcaption></figure>

*Proof.* The only allowed multiplicity sequences that are subsets of [[3, 2[⁷]]] are [2[⁴]], [3], and [3, 2]; these are all simple singularities of types $A_{2k}$, $E_6$, and $E_8$, respectively; their links are $T(2, 2k + 1)$, $T(3, 4)$, and $T(3, 5)$, respectively.

Suppose such a curve exists; perturb it to get a smooth curve $C'$ of genus 10, and take the double cover $\Sigma(\mathbb{CP}^2, C')$ of $\mathbb{CP}^2$ branched over $C'$; this is a $K3$ surface, and, by construction, it contains (as pairwise disjointly embedded) the double covers of $\mathbb{B}^4$ branched over the Milnor fiber of all the singularities of $C$.

All these Milnor fibers are negative definite, because the singularities are simple, and their second Betti numbers sum to 20, which contradicts the fact that $b_2^-(K3) = 19$. $\square$

This is where our approach for sextics differs from the previous cases we examined: here we are not trying to classify fillings, but rather looking specifically at embeddings in $\mathbb{CP}^2$ or, equivalently, at rational homology ball fillings. Moreover, the same argument can be used to narrow down the number of different splittings of the multiplicity multi-sequences that one needs to consider; namely, there can be no sextic whose singularities are all simple.

*Remark 7.14.* A similar argument to that of the proposition above, using five-fold covers instead of two-fold covers, can be used to obstruct the existence of the two quintics in $\mathbb{CP}^2$ with five and six singularities (which we excluded in Example 3.20; the non-fillability of the corresponding cuspidal contact structures was established in Proposition 7.7). A general result, of a more topological flavor, has been proved in [GK22, Theorem 4.7].

## 8. Differences between the complex and symplectic categories

Every complex algebraic curve in $\mathbb{CP}^2$ is a symplectic surface (potentially singular), but most symplectic surfaces are not complex algebraic. However, in many situations, a symplectic surface is symplectically isotopic to a complex curve. This is known for smooth symplectic surfaces of degree at most 17 and is conjectured for smooth symplectic surfaces in $\mathbb{CP}^2$ in general. However,
<figure><img src="image_49.png" /><figcaption>FIGURE 42. The fake Pappus configuration.</figcaption></figure>

for singular curves, there need not always be an equisingular symplectic isotopy from a symplectic configuration to a complex configuration. This is easiest to verify when there is a singular symplectic configuration whose singularity types cannot be realized by any complex curve configuration. Some examples of reducible configurations with this property come from surprising and ancient theorems in projective geometry. For example, the Pappus theorem shows that given a configuration of lines as in Figure 42, the points X, Y, Z are necessarily collinear. Therefore, there is no complex line arrangement consisting of the lines in Figure 42 together with an extra line passing through X and Y but not Z.

Other examples of reducible symplectic configurations which are not realizable with complex algebraic curves come from pseudoline arrangements. A pseudoline arrangement is a collection of simple closed curves in $\mathbb{RP}^2$ each of which represents the homology class $[\mathbb{RP}^1]$. It was proven in [RS19] that any pseudoline arrangement in $\mathbb{RP}^2$ can be extended (after isotopy) to a symplectic line arrangement in $\mathbb{CP}^2$. Therefore, any pseudoline arrangement which is not realizable by straight lines (over the complex numbers) gives another example of this phenomenon.

We have seen for rational cuspidal curves in the unicuspidal and low-degree cases we looked at, every symplectic realization in $\mathbb{CP}^2$ is symplectically isotopic to a complex curve. This is in contrast to the differences appearing in the reducible configurations just mentioned. However, the differences between symplectic and complex singular curves are not restricted to reducible configurations.

Using birational transformations, we provide here an example of an irreducible singular curve which is realizable symplectically but not complex algebraically. This example was given to us by Orevkov, and we are very grateful to him for explaining it to us. Although this curve is irreducible, its singularities are not cuspidal, they are locally reducible. It remains open as to whether there are any symplectic rational cuspidal curves in $\mathbb{CP}^2$ which are not symplectically isotopic to a complex rational cuspidal curve.

**THEOREM 8.1.** *There is an irreducible symplectic rational curve of degree eight in $\mathbb{CP}^2$ that is not equisingularly isotopic to a complex curve.*

More precisely, we demonstrate a specific example of a symplectic surface $C$ of degree eight, with three *reducible* singularities (not cuspidal), each isomorphic to a singularity of type (3,5) plus a generic line. That is, the singularity is defined by an equation locally modeled on $x(x^3 + y^5) = 0$. We then prove that no complex curve with the same singularities can exist.
<figure><img src="image_49.png" /><figcaption>FIGURE 42. The fake Pappus configuration.</figcaption></figure>

for singular curves, there need not always be an equisingular symplectic isotopy from a symplectic configuration to a complex configuration. This is easiest to verify when there is a singular symplectic configuration whose singularity types cannot be realized by any complex curve configuration. Some examples of reducible configurations with this property come from surprising and ancient theorems in projective geometry. For example, the Pappus theorem shows that given a configuration of lines as in Figure 42, the points X, Y, Z are necessarily collinear. Therefore, there is no complex line arrangement consisting of the lines in Figure 42 together with an extra line passing through X and Y but not Z.

Other examples of reducible symplectic configurations which are not realizable with complex algebraic curves come from pseudoline arrangements. A pseudoline arrangement is a collection of simple closed curves in $\mathbb{RP}^2$ each of which represents the homology class $[\mathbb{RP}^1]$. It was proven in [RS19] that any pseudoline arrangement in $\mathbb{RP}^2$ can be extended (after isotopy) to a symplectic line arrangement in $\mathbb{CP}^2$. Therefore, any pseudoline arrangement which is not realizable by straight lines (over the complex numbers) gives another example of this phenomenon.

We have seen for rational cuspidal curves in the unicuspidal and low-degree cases we looked at, every symplectic realization in $\mathbb{CP}^2$ is symplectically isotopic to a complex curve. This is in contrast to the differences appearing in the reducible configurations just mentioned. However, the differences between symplectic and complex singular curves are not restricted to reducible configurations.

Using birational transformations, we provide here an example of an irreducible singular curve which is realizable symplectically but not complex algebraically. This example was given to us by Orevkov, and we are very grateful to him for explaining it to us. Although this curve is irreducible, its singularities are not cuspidal, they are locally reducible. It remains open as to whether there are any symplectic rational cuspidal curves in $\mathbb{CP}^2$ which are not symplectically isotopic to a complex rational cuspidal curve.

**THEOREM 8.1.** *There is an irreducible symplectic rational curve of degree eight in $\mathbb{CP}^2$ that is not equisingularly isotopic to a complex curve.*

More precisely, we demonstrate a specific example of a symplectic surface $C$ of degree eight, with three *reducible* singularities (not cuspidal), each isomorphic to a singularity of type (3,5) plus a generic line. That is, the singularity is defined by an equation locally modeled on $x(x^3 + y^5) = 0$. We then prove that no complex curve with the same singularities can exist.
<figure><img src="image_50.png" /></figure>

FIGURE 43. On the left is a degree-eight curve with three reducible singularities with one branch of type (3,5) and the other smooth. A triangle of lines passes through these three singular points. On the right is the same triangle of lines with three conics. At each intersection point on the triangle, one pair of the three conics intersects tangentially and the third intersects these two transversally. Configuration $\mathcal{D}$ is built by overlaying the two sides of this configuration. The three lines on each side coincide and the tangent direction to the (3,5)-cusp part of each singularity of the curve agrees with the common tangent direction to two of the three conics at the intersection points.

To understand the symplectic existence and complex obstruction to this rational singular curve, we relate it via a birational transformation to a reducible configuration that is easier to understand.

Define a configuration $\mathcal{D}$ by adding to the degree-eight rational singular curve $C$, three conics and three lines with the following intersection conditions. The three lines $\ell_1, \ell_2, \ell_3$ must pass through the three pairs of the singular points of $C$. The three conics $q_1, q_2, q_3$ must each pass through the three singular points of $C$, and each must be simply tangent to two of the cuspidal branches and transverse to the third (none should be tangent to the smooth branch of $C$ at the singularities or the lines $\ell_i$). The singular points of $C$ are the only singular points of $\mathcal{D}$ because there are no further intersections between $C, \ell_1, \ell_2, \ell_3, q_1, q_2, q_3$ for degree reasons. (See Figure 43.)

Define another configuration $\mathcal{T}$, built from a triangle of three lines with an inscribed conic, and three *non-concurrent* lines drawn from a vertex of the triangle to the opposite tangency point. (See Figure 44.)

LEMMA 8.2. The two configurations $\mathcal{D}$ and $\mathcal{T}$ are birationally equivalent.

*Proof.* Blow up at the singularities of $C$, creating three exceptional divisors $e_1, e_2, e_3$. The singularities of the proper transform $\tilde{C}$ are three simple cusps, and for each cusp, there is one exceptional divisor $e_i$ intersecting $\tilde{C}$ tangentially at that cusp and transversally at one other smooth point of $\tilde{C}$. Each $\ell_i$ has been blown up twice, so the self-intersection number of each $\tilde{\ell}_i$ is $-1$. Therefore, we can blow down the proper transforms $\tilde{\ell}_1, \tilde{\ell}_2, \tilde{\ell}_3$, returning to $\mathbb{CP}^2$ with a new configuration of curves. Here $\tilde{C}$ blows down to a quartic $\mathcal{D}$ with three simple cusp singularities, and $q_1, q_2, q_3$ blow down to three lines, each passing through two of the simple cusps. The three exceptional divisors $e_1, e_2, e_3$ blow down to three tangent lines to the cusps of $\mathcal{D}$, and they intersect in three distinct points (i.e. the images of $\tilde{\ell}_1, \tilde{\ell}_2, \tilde{\ell}_3$). (See Figure 45.)

Now blow up at the simple cusps of $\mathcal{D}$, resolving the singularities and creating three tangent exceptional divisors $f_1, f_2, f_3$. The proper transforms of the lines $q_1, q_2, q_3$ become exceptional divisors which can then be blown down. The result is the configuration $\mathcal{T}$ where $\mathcal{D}$ is sent to the smooth conic $Q, f_1, f_2, f_3$ become its tangent lines, and $e_1, e_2, e_3$ blow down to lines, each
<figure><img src="image_50.png" /></figure>

FIGURE 43. On the left is a degree-eight curve with three reducible singularities with one branch of type (3,5) and the other smooth. A triangle of lines passes through these three singular points. On the right is the same triangle of lines with three conics. At each intersection point on the triangle, one pair of the three conics intersects tangentially and the third intersects these two transversally. Configuration $\mathcal{D}$ is built by overlaying the two sides of this configuration. The three lines on each side coincide and the tangent direction to the (3,5)-cusp part of each singularity of the curve agrees with the common tangent direction to two of the three conics at the intersection points.

To understand the symplectic existence and complex obstruction to this rational singular curve, we relate it via a birational transformation to a reducible configuration that is easier to understand.

Define a configuration $\mathcal{D}$ by adding to the degree-eight rational singular curve $C$, three conics and three lines with the following intersection conditions. The three lines $\ell_1, \ell_2, \ell_3$ must pass through the three pairs of the singular points of $C$. The three conics $q_1, q_2, q_3$ must each pass through the three singular points of $C$, and each must be simply tangent to two of the cuspidal branches and transverse to the third (none should be tangent to the smooth branch of $C$ at the singularities or the lines $\ell_i$). The singular points of $C$ are the only singular points of $\mathcal{D}$ because there are no further intersections between $C, \ell_1, \ell_2, \ell_3, q_1, q_2, q_3$ for degree reasons. (See Figure 43.)

Define another configuration $\mathcal{T}$, built from a triangle of three lines with an inscribed conic, and three *non-concurrent* lines drawn from a vertex of the triangle to the opposite tangency point. (See Figure 44.)

LEMMA 8.2. The two configurations $\mathcal{D}$ and $\mathcal{T}$ are birationally equivalent.

*Proof.* Blow up at the singularities of $C$, creating three exceptional divisors $e_1, e_2, e_3$. The singularities of the proper transform $\tilde{C}$ are three simple cusps, and for each cusp, there is one exceptional divisor $e_i$ intersecting $\tilde{C}$ tangentially at that cusp and transversally at one other smooth point of $\tilde{C}$. Each $\ell_i$ has been blown up twice, so the self-intersection number of each $\tilde{\ell}_i$ is $-1$. Therefore, we can blow down the proper transforms $\tilde{\ell}_1, \tilde{\ell}_2, \tilde{\ell}_3$, returning to $\mathbb{CP}^2$ with a new configuration of curves. Here $\tilde{C}$ blows down to a quartic $\mathcal{D}$ with three simple cusp singularities, and $q_1, q_2, q_3$ blow down to three lines, each passing through two of the simple cusps. The three exceptional divisors $e_1, e_2, e_3$ blow down to three tangent lines to the cusps of $\mathcal{D}$, and they intersect in three distinct points (i.e. the images of $\tilde{\ell}_1, \tilde{\ell}_2, \tilde{\ell}_3$). (See Figure 45.)

Now blow up at the simple cusps of $\mathcal{D}$, resolving the singularities and creating three tangent exceptional divisors $f_1, f_2, f_3$. The proper transforms of the lines $q_1, q_2, q_3$ become exceptional divisors which can then be blown down. The result is the configuration $\mathcal{T}$ where $\mathcal{D}$ is sent to the smooth conic $Q, f_1, f_2, f_3$ become its tangent lines, and $e_1, e_2, e_3$ blow down to lines, each
<figure><img src="image_51.png" /><figcaption>FIGURE 44. Configuration $\mathcal{T}$ consists of a conic with three tangent lines, together with three additional lines passing through the tangent intersections and the intersection of the other two lines, and intersecting each other in double points as shown. In a complex algebraic arrangement, these three additional lines would necessarily intersect each other all at the same point in a triple intersection.</figcaption></figure>

<figure><img src="image_52.png" /><figcaption>FIGURE 45. Intermediate configuration between configurations $\mathcal{D}$ and $\mathcal{T}$. The singular component is a degree-four curve with three simple cusps. There is one line through each pair of these three cusps, together with one line tangent to each cusp. The intersections of the three tangent lines are required to be three distinct double points instead of a coinciding triple point.</figcaption></figure>

connecting a tangency of $Q$ and $f_i$ with the intersection of $f_{i+1}$ and $f_{i-1}$ (here we consider the labels modulo three). $\square$

**LEMMA 8.3.** *The configuration $\mathcal{T}$ is symplectically realizable, but not complex realizable.*

*Proof.* We start by proving that $\mathcal{T}$ is not complex realizable. In fact, this follows from Brianchon's theorem: this states that if an hexagon $ABCDEF$ is circumscribed to a conic, then $AD, BE$, and $CF$ are concurrent. Indeed, if we call $ACE$ the vertices of the triangle in $T$, and $B, D, F$ the tangency points of the conics (so that $B$ lies on $AC$), then $ABCDEF$ is a degenerate hexagon circumscribed to a conic, and therefore $AD, BF$, and $CE$ are concurrent.

To prove the existence of a symplectic realization of $\mathcal{T}$, we just perturb the degenerate Brianchon configuration locally around the triple point of intersection of the secants. As the symplectic condition is open, performing this operation in a $C^1$-small way preserves the fact that the curves are symplectic. $\square$
<figure><img src="image_51.png" /></figure>

FIGURE 44. Configuration $\mathcal{T}$ consists of a conic with three tangent lines, together with three additional lines passing through the tangent intersections and the intersection of the other two lines, and intersecting each other in double points as shown. In a complex algebraic arrangement, these three additional lines would necessarily intersect each other all at the same point in a triple intersection.

<figure><img src="image_52.png" /></figure>

FIGURE 45. Intermediate configuration between configurations $\mathcal{D}$ and $\mathcal{T}$. The singular component is a degree-four curve with three simple cusps. There is one line through each pair of these three cusps, together with one line tangent to each cusp. The intersections of the three tangent lines are required to be three distinct double points instead of a coinciding triple point.

connecting a tangency of $Q$ and $f_i$ with the intersection of $f_{i+1}$ and $f_{i-1}$ (here we consider the labels modulo three). $\square$

LEMMA 8.3. The configuration $\mathcal{T}$ is symplectically realizable, but not complex realizable.

*Proof.* We start by proving that $\mathcal{T}$ is not complex realizable. In fact, this follows from Brianchon's theorem: this states that if an hexagon $ABCDEF$ is circumscribed to a conic, then $AD, BE$, and $CF$ are concurrent. Indeed, if we call $ACE$ the vertices of the triangle in $T$, and $B, D, F$ the tangency points of the conics (so that $B$ lies on $AC$), then $ABCDEF$ is a degenerate hexagon circumscribed to a conic, and therefore $AD, BF$, and $CE$ are concurrent.

To prove the existence of a symplectic realization of $\mathcal{T}$, we just perturb the degenerate Brianchon configuration locally around the triple point of intersection of the secants. As the symplectic condition is open, performing this operation in a $C^1$-small way preserves the fact that the curves are symplectic. $\square$
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL cuspidal CURVES

Proof of Theorem 8.1. We construct the curve *C* starting from the configuration *T*; indeed, Lemma 8.3 shows that *T* exists symplectically, and by Lemma 8.2, *D* is birationally equiv- alent to it; because *C* is a component of *D*, we have proved its existence as a symplectic curve.

Conversely, suppose that $C$ existed as a complex curve. We can augment any such realization of $C$ to a configuration $\mathcal{D}$ as follows. Let $p_1, p_2, p_3$ be the three singular points of $C$. There is a unique complex line through any pair of distinct points, so let $\ell_1, \ell_2, \ell_3$ be the unique lines through the three pairs of $p_1, p_2, p_3$. Each transverse intersection of $\ell_i$ with $C$ at $p_j$ contributes +4 to their intersection number. Therefore, $p_1, p_2, p_3$ cannot be collinear because then $C$ would intersect a line with intersection number 12 instead of 8. Therefore, the three lines $\ell_i$ are distinct. Furthermore, the intersections of $\ell_i$ with $C$ must be transverse, because a tangential intersection would add an additional positive intersection beyond the required eight. Similarly, the $\ell_i$ cannot intersect $C$ at any other point besides the $p_j$.

If we fix two distinct points $A, B \in \mathbb{CP}^2$ with complex lines in their tangent spaces $T_A, T_B$, and a third distinct point $C \in \mathbb{CP}^2$, there is a unique complex conic through $A, B$, and $C$ tangent to $T_A$ and $T_B$. (These five conditions give five linear constraints on the six projective coefficients of the six degree-two monomials. The solution is unique because any two such conics would have intersection number five instead of four.) In general, it is possible for this conic to be singular by degenerating into two lines. Let $T_1, T_2$ and $T_3$ be the complex tangent line directions to the simple cusps of $C$ at $p_1, p_2$, and $p_3$. Letting $A = p_i, T_A = T_i, B = p_{i+1}, T_B = T_{i+1}$, and $C = p_{i+2}$ with mod 3 cyclic indices for $i = 1, 2, 3$, we construct three conics as required in the configuration $\mathcal{D}$. To check that none of these conics is a degenerate pair of lines, we observe that the unique pair of lines through $AB$ and $AC$ is a pair of the $\ell_i$ which are not tangent to $T_A$ and, thus, cannot be the chosen conic.

Therefore, if we had a complex realization of $C$, we could construct a complex realization of the configuration $\mathcal{D}$ and then use the birational transformation to construct a complex realization of the configuration $\mathcal{T}$, but this is impossible. $\square$

ACKNOWLEDGEMENTS

This project originated from conversations the two authors had during the intensive semester
on *Symplectic geometry and topology* at the Mittag-Leffler Institute; we acknowledge their
hospitality and the great working environment. Further progress was made during the BIRS con-
ference *Thirty years of Floer theory for 3-manifolds* in Oaxaca; we thank the organizers for this
excellent conference and for the opportunity to collaborate. The authors would like to thank the
participants of the AIM workshop *Symplectic four-manifolds through branched coverings* for their
interest in the project. MG acknowledges hospitality from Stanford University and UC Davis. LS
is grateful for hospitality from Université de Nantes. We would like to thank Józsi Bodnár, Erwan
Brugallé, Roger Casals, Anthony Conway, Paolo Ghiggini, András Némethi, Tomasz Pełka, Olga
Plamenevskaya, Danny Ruberman, András Stipsicz, and Chris Wendl. Finally, the authors would
like to thank Stepan Orevkov for his comments and for communicating to us the example
in § 8.

**Appendix A. Symplectic rational ball fillings of lens spaces**

In this section, we prove two results on rational homology ball fillings of lens spaces. The first
has recently appeared in independent work of Fossati [Fos20, Theorem 4].
THE SYMPLECTIC ISOTOPY PROBLEM FOR RATIONAL CUSPIDAL CURVES

Proof of Theorem 8.1. We construct the curve *C* starting from the configuration *T*; indeed, Lemma 8.3 shows that *T* exists symplectically, and by Lemma 8.2, *D* is birationally equiv- alent to it; because *C* is a component of *D*, we have proved its existence as a symplectic curve.

Conversely, suppose that $C$ existed as a complex curve. We can augment any such realization of $C$ to a configuration $\mathcal{D}$ as follows. Let $p_1, p_2, p_3$ be the three singular points of $C$. There is a unique complex line through any pair of distinct points, so let $\ell_1, \ell_2, \ell_3$ be the unique lines through the three pairs of $p_1, p_2, p_3$. Each transverse intersection of $\ell_i$ with $C$ at $p_j$ contributes +4 to their intersection number. Therefore, $p_1, p_2, p_3$ cannot be collinear because then $C$ would intersect a line with intersection number 12 instead of 8. Therefore, the three lines $\ell_i$ are distinct. Furthermore, the intersections of $\ell_i$ with $C$ must be transverse, because a tangential intersection would add an additional positive intersection beyond the required eight. Similarly, the $\ell_i$ cannot intersect $C$ at any other point besides the $p_j$.

If we fix two distinct points $A, B \in \mathbb{CP}^2$ with complex lines in their tangent spaces $T_A, T_B$, and a third distinct point $C \in \mathbb{CP}^2$, there is a unique complex conic through $A, B$, and $C$ tangent to $T_A$ and $T_B$. (These five conditions give five linear constraints on the six projective coefficients of the six degree-two monomials. The solution is unique because any two such conics would have intersection number five instead of four.) In general, it is possible for this conic to be singular by degenerating into two lines. Let $T_1, T_2$ and $T_3$ be the complex tangent line directions to the simple cusps of $C$ at $p_1, p_2$, and $p_3$. Letting $A = p_i, T_A = T_i, B = p_{i+1}, T_B = T_{i+1}$, and $C = p_{i+2}$ with mod 3 cyclic indices for $i = 1, 2, 3$, we construct three conics as required in the configuration $\mathcal{D}$. To check that none of these conics is a degenerate pair of lines, we observe that the unique pair of lines through $AB$ and $AC$ is a pair of the $\ell_i$ which are not tangent to $T_A$ and, thus, cannot be the chosen conic.

Therefore, if we had a complex realization of $C$, we could construct a complex realization of the configuration $\mathcal{D}$ and then use the birational transformation to construct a complex realization of the configuration $\mathcal{T}$, but this is impossible. □

ACKNOWLEDGEMENTS

This project originated from conversations the two authors had during the intensive semester
on *Symplectic geometry and topology* at the Mittag-Leffler Institute; we acknowledge their
hospitality and the great working environment. Further progress was made during the BIRS con-
ference *Thirty years of Floer theory for 3-manifolds* in Oaxaca; we thank the organizers for this
excellent conference and for the opportunity to collaborate. The authors would like to thank the
participants of the AIM workshop *Symplectic four-manifolds through branched coverings* for their
interest in the project. MG acknowledges hospitality from Stanford University and UC Davis. LS
is grateful for hospitality from Université de Nantes. We would like to thank Józsi Bodnár, Erwan
Brugallé, Roger Casals, Anthony Conway, Paolo Ghiggini, András Némethi, Tomasz Pełka, Olga
Plamenevskaya, Danny Ruberman, András Stipsicz, and Chris Wendl. Finally, the authors would
like to thank Stepan Orevkov for his comments and for communicating to us the example
in § 8.

**Appendix A. Symplectic rational ball fillings of lens spaces**

In this section, we prove two results on rational homology ball fillings of lens spaces. The first
has recently appeared in independent work of Fossati [Fos20, Theorem 4].
**PROPOSITION A.1.** If a lens space $(L(p, q), \xi)$ has a weak symplectic rational homology ball filling, then there exist coprime integers $0 < k < m$ such that $(p, q) = (m^2, mk - 1)$, and $\xi$ is universally tight.

The family coincides with the family of cyclic quotient singularities with rational disk smooth-
ing, which were classified by Wahl [Wah81, Example 5.9.1] (see also [Lis08, Corollary 1.2]).
In the proof, we use the Ozsváth–Szabó contact invariant [OSz05] in Heegaard Floer homol-
ogy [OSz04b]; the relevant properties of the theory are the non-vanishing of the contact invariant
for fillable contact structures [OSz05, Theorem 1.5], the fact that lens spaces are L-spaces (i.e.
they have the smallest possible Heegaard Floer homology) [OSz04a, Proposition 3.1], and the
absolute grading on Heegaard Floer homology [OSz03].

*Proof.* Call $L = L(p, q)$ and $W$ a weak symplectic rational homology ball filling of $(L, \xi)$. As $\xi$ is fillable, it is tight [EG91]. All tight contact structures on lens spaces are planar [Sch07], and it follows from Wendl's theorem that all weak symplectic fillings can be deformed to Stein [Wen10]. Let $J$ denote a complex structure such that $(W, J)$ is a Stein filling of $(L, \xi)$.

In particular, $W$ admits a handle decompositions with no 3-handles. This implies that $\pi_1(W)$ is a quotient of $\pi_1(L)$, and hence it is cyclic, and that $H_1(W, L) \cong H^3(W) = 0$. As $L$ bounds a rational ball, $p = m^2$ for some positive integer $m$, and from the long exact sequence of the pair $(W, L)$ and the universal coefficient theorem, it follows that $H_1(W) = \mathbb{Z}/m\mathbb{Z}$ (this is classical; see, e.g., [AG17, Proposition 2.2]).

Let $W'$ be the universal cover of $W$. We know that $\chi(W') = m\chi(W) = m$. As $W'$ is simply-connected, $H_1(W') = 0$ and $H^2(W')$ is torsion-free. As $W'$ has a decomposition without 3-handles, lifted from that of $W$, we deduce that $H_3(W') = 0$ and that $H_2(W')$ is torsion-free. Adding all pieces together, it follows that $H_2(W') = \mathbb{Z}^{m-1}$. The boundary $L'$ is the $m$-fold cover of $L$ and, therefore, $L' = L(m, q)$. Let $\xi'$ be the pull-back of $\xi$ to $L$, filled by the pull-back $J'$ of $J$. The first Chern class $c_1(J')$ of $J'$ is the pull-back under the covering map of $c_1(J)$. As the latter is torsion and $H^2(W')$ is torsion-free, $c_1(J') = 0$.

The Ozsváth–Szabó contact invariant $c(\xi') \in \text{HF}^+(\mathbb{L}', \mathbb{t}_{\xi'})$ lives in degree $d(\mathbb{L}', \mathbb{t}_{\xi'})$, computed as the degree of the map associated to the cobordism $X = W' \setminus B^4$. The degree of the map $F_{X, \mathbb{L}', \mathbb{t}_{\xi'}}$ carrying $c(\xi)$ to $c(\xi_{\text{st}})$ is

$$
\deg F_{X, \mathbb{L}', \mathbb{t}_{\xi'}} = \frac{c_1^2(\mathbb{L}') - 2\chi(X) - 3\sigma(X)}{4} = \frac{m-1}{4}
$$

and, hence, $d(-L', t_{\xi'}) = (m-1)/4$.

It follows from [AG17, Lemma 4.5] that $q \equiv \pm 1 \pmod m$. By explicitly computing correction terms of $L(m, \pm 1)$, we see that if $q \equiv 1 \pmod m$ there is no spin $c$ structure on $L(m, q)$ with correction term $(m-1)/4$.

Therefore, we reduced to the case when $q \equiv -1 \pmod m$. That is, $q = mk - 1$ for some $0 < k \le m$. As $L$ bounds a smooth rational homology ball, it follows from Lisca's classification of lens spaces that bound rational homology balls that $\gcd(m, k) = 1$ or $\gcd(m, k) = 2$ [Lis07] (see [Lec12, p. 247] for an amendment to the statement of [Lis07, Theorem 1.2] that includes the case $\gcd(m, k) = 2$). The former case corresponds to structures in Wahl's family, and we set out to exclude the second.

In this case, $m = 2m''$ is even, and we can look at the $m''$-fold cover $(W'', J'')$ of $(W, J)$, with boundary $(L'', \xi'')$. Observe that our assumption $\gcd(m, k) = 2$ implies that $L'' = L(2m, 2m-1)$. The same argument as before shows that $d(-L'', t_{\xi''}) = (m'' - 1)/4 = (m - 2)/8$. However, by classification of tight contact structures on lens spaces [Hon00], $L(2m, 2m-1)$ admits a unique
PROPOSITION A.1. If a lens space $(L(p, q), \xi)$ has a weak symplectic rational homology ball filling, then there exist coprime integers $0 < k < m$ such that $(p, q) = (m^2, mk - 1)$, and $\xi$ is universally tight.

The family coincides with the family of cyclic quotient singularities with rational disk smooth- ing, which were classified by Wahl [Wah81, Example 5.9.1] (see also [Lis08, Corollary 1.2]). In the proof, we use the Ozsváth–Szabó contact invariant [OSz05] in Heegaard Floer homol- ogy [OSz04b]; the relevant properties of the theory are the non-vanishing of the contact invariant for fillable contact structures [OSz05, Theorem 1.5], the fact that lens spaces are L-spaces (i.e. they have the smallest possible Heegaard Floer homology) [OSz04a, Proposition 3.1], and the absolute grading on Heegaard Floer homology [OSz03].

*Proof.* Call $L = L(p, q)$ and $W$ a weak symplectic rational homology ball filling of $(L, \xi)$. As $\xi$ is fillable, it is tight [EG91]. All tight contact structures on lens spaces are planar [Sch07], and it follows from Wendl’s theorem that all weak symplectic fillings can be deformed to Stein [Wen10]. Let $J$ denote a complex structure such that $(W, J)$ is a Stein filling of $(L, \xi)$.

In particular, $W$ admits a handle decompositions with no 3-handles. This implies that $\pi_1(W)$ is a quotient of $\pi_1(L)$, and hence it is cyclic, and that $H_1(W, L) \cong H^3(W) = 0$. As $L$ bounds a rational ball, $p = m^2$ for some positive integer $m$, and from the long exact sequence of the pair $(W, L)$ and the universal coefficient theorem, it follows that $H_1(W) = \mathbb{Z}/m\mathbb{Z}$ (this is classical; see, e.g., [AG17, Proposition 2.2]).

Let $W'$ be the universal cover of $W$. We know that $\chi(W') = m\chi(W) = m$. As $W'$ is simply-connected, $H_1(W') = 0$ and $H^2(W')$ is torsion-free. As $W'$ has a decomposition without 3-handles, lifted from that of $W$, we deduce that $H_3(W') = 0$ and that $H_2(W')$ is torsion-free. Adding all pieces together, it follows that $H_2(W') = \mathbb{Z}^{m-1}$. The boundary $L'$ is the $m$-fold cover of $L$ and, therefore, $L' = L(m, q)$. Let $\xi'$ be the pull-back of $\xi$ to $L$, filled by the pull-back $J'$ of $J$. The first Chern class $c_1(J')$ of $J'$ is the pull-back under the covering map of $c_1(J)$. As the latter is torsion and $H^2(W')$ is torsion-free, $c_1(J') = 0$.

The Ozsváth–Szabó contact invariant $c(\xi') \in \text{HF}^+(\mathbb{Z}^m, \mathbb{Z})$ lives in degree $d(\mathbb{Z}^m, \mathbb{Z})$, computed as the degree of the map associated to the cobordism $X = W' \setminus B^4$. The degree of the map $F_{X, \mathbb{Z}}$ carrying $c(\xi)$ to $c(\xi_{\text{st}})$ is

$$ \deg F_{X, \mathbb{Z}} = \frac{c_1^2(J') - 2\chi(X) - 3\sigma(X)}{4} = \frac{m-1}{4} $$

and, hence, $d(-L', t_{\xi'}) = (m-1)/4$.

It follows from [AG17, Lemma 4.5] that $q \equiv \pm 1 \pmod m$. By explicitly computing correction terms of $L(m, \pm 1)$, we see that if $q \equiv 1 \pmod m$ there is no spin $c$ structure on $L(m, q)$ with correction term $(m-1)/4$.

Therefore, we reduced to the case when $q \equiv -1 \pmod m$. That is, $q = mk - 1$ for some $0 < k \le m$. As $L$ bounds a smooth rational homology ball, it follows from Lisca’s classification of lens spaces that bound rational homology balls that $\text{gcd}(m, k) = 1$ or $\text{gcd}(m, k) = 2$ [Lis07] (see [Lec12, p. 247] for an amendment to the statement of [Lis07, Theorem 1.2] that includes the case $\text{gcd}(m, k) = 2$). The former case corresponds to structures in Wahl’s family, and we set out to exclude the second.

In this case, $m = 2m''$ is even, and we can look at the $m''$-fold cover $(W'', J'')$ of $(W, J)$, with boundary $(L'', \xi'')$. Observe that our assumption $\text{gcd}(m, k) = 2$ implies that $L'' = L(2m, 2m - 1)$. The same argument as before shows that $d(-L'', t_{\xi''}) = (m'' - 1)/4 = (m - 2)/8$. However, by classification of tight contact structures on lens spaces [Hon00], $L(2m, 2m - 1)$ admits a unique
tight contact structure $\xi_0$, namely the boundary of the plumbing of $2m - 1$ copies of $T^*S^2$, whose corresponding contact structure has deg $c(\xi_0) = (-2m + 1)/4$.

To conclude the proof, we need to show that $\xi'$ is universally tight. As mentioned previously, however, $\xi'$ is tight (because it is filled by $W'$), and $L' = L(m, m-1)$ admits a unique tight contact structure (by [Hon00]). As on each lens space there is always a universally tight contact structure (obtained by taking a quotient of $S^3$ with its standard contact structure), $\xi'$ is universally tight. $\square$

Alternatively, to exclude the case $\gcd(k, m) = 2$ in the proof, one can argue that the unique contact structure on $L(2m, 2m-1)$ is also universally tight, and then so is $\xi$. Then Lisca's classification of fillings [Lis08, Corollary 1.2] rules this possibility out.

The second result concerns the uniqueness of the rational homology ball filling. It is very likely that the result is known to experts (either in the context of Milnor fibers of cyclic quotient singularities, or in the context of symplectic fillings of contact structures), but we were unable to locate the precise statement.

**PROPOSITION A.2.** The standard contact structure on the lens space $L(m^2, mk-1)$ has a unique rational homology ball symplectic filling up to symplectic deformation.

The proof is, in fact, somewhat implicit in Lisca's classification paper [Lis08]. We refer to the paper for notation and context. We start by recalling the following two facts about continued fractions. Both facts follow from Riemenschneier's point rule [Rie74] (see [Lis08, Lemma 2.6]).

If we have two continued fraction expansions $[\mathbf{a}]^{-}$ and $[\mathbf{b}]^{-}$ such that

$$1/[a_1, \dots, a_m]^{-} + 1/[b_1, \dots, b_\ell]^{-} = 1$$

(or, equivalently, the two associated fractions are of the form $p/q$ and $p/(p-q)$ for some $q < p$ coprime positive integers), we say that $[\mathbf{a}]^{-}$ and $[\mathbf{b}]^{-}$ are dual to each other.

If $[\mathbf{a}]^{-}$ and $[\mathbf{b}]^{-}$ are dual to each other, then $\sum (a_i - 1) = \sum (b_j - 1)$. We also have that

$$[n_1, \dots, n_{j-1}, 1, n_{j+1}, \dots, n_\ell]^{-} = 0$$

if and only if $[n_{j-1}, \dots, n_2, n_1]^{-}$ and $[n_{j+1}, \dots, n_\ell]^{-}$ are dual to each other.

*Proof.* The (negative) continued fraction expansion of $m^2/(mk-1)$ is of the form $[n_1, \dots, n_{j-1}, 2, n_{j+1}, \dots, n_\ell]^{-}$, such that $[n_1, \dots, n_{j-1}, 1, n_{j+1}, \dots, n_\ell]^{-} = 0$ (see the proof of [Lis08, Theorem 1.2(c)]).

Recall from the proof of [Lis08, Theorem 1.1] that symplectic deformation classes of symplectic fillings of $L(p, q)$ with $p/q = [n_1, \dots, n_\ell]^{-}$ correspond to strings $\mathbf{m} = (m_1, \dots, m_\ell)$ such that $1 \le m_i \le n_i$ for each $i$ and $[m_1, \dots, m_\ell]^{-} = 0$.

Moreover, the Euler characteristics of the filling corresponding to $\mathbf{m}$ is $\sum (n_i - m_i)$, so that the filling is a rational homology ball if and only if $m_i = n_i$ for all indices except for one, for which $m_i = n_i - 1$. We claim that for such any $p, q$ there is at most one index $j$ such that the corresponding sequence $\mathbf{m} = (n_1, \dots, n_{j-1}, n_j - 1, n_{j+1}, n_\ell)$ has $[\mathbf{m}]^{-} = 0$, and in that case $n_j = 2$.

If $n_j > 2$, then $[\mathbf{m}]^{-}$ is the continued fraction expansion of a rational number $p'/q' > 1$. Suppose now $n_j = 2$. If $[\mathbf{m}]^{-} = 0$, then $[n_{j-1}, \dots, n_1]^{-}$ and $[n_{j+1}, \dots, n_\ell]^{-}$ are dual to each other; however, as $j$ varies between 1 and $\ell$, the difference

$$\sum_{h<j} (n_h - 1) - \sum_{h>J} (n_h - 1)$$