An explicit example of such principal terms arguments done at the level of Gromov-Witten classes can be found in [22], where it is shown that relative Gromov-Witten classes of $\mathbb{P}^1$ are tautological.

(iv) There is a version of Theorem 5.3 for Gromov-Witten classes: under the assumptions of Theorem 5.3, the nodal Gromov-Witten classes of $X$ can be effectively reconstructed from the Gromov-Witten classes of $X_1, X_2, D$, and $Z$, and are tautological if the Gromov-Witten classes of $X_1, X_2, D$, and $Z$ are tautological. Indeed, the degeneration formula of Theorem 2.3 is stated at the level of virtual classes on moduli space of stable maps and so can be applied to study Gromov-Witten classes. In addition, the blow-up result of [24] also admits a version for Gromov-Witten classes preserving the property of being tautological because its proof consists of principal terms arguments based on degeneration and localization formulas, as the results of [60] mentioned in (iii).

The proof using (i)-(iv), in fact, provides an algorithm to calculate the Gromov-Witten classes of complete intersections. ♦

*Remark 5.5.* Using Theorem 5.4, we can give a slightly different proof of the result proved in [41] that the Gromov-Witten classes of curves are tautological. Indeed, by degeneration, it is enough to prove that the relative Gromov-Witten classes of a genus 1 curve are tautological. By the results of [60] expressing relative Gromov-Witten theory in terms of absolute, and lifted to the level of Gromov-Witten classes as discussed in (iii) of the proof of Theorem 5.4, it is enough to show that the Gromov-Witten classes of a genus 1 curve are tautological. But this follows from Theorem 5.4 because a genus 1 curve is a cubic in $\mathbb{P}^2$.

## APPENDIX A. NODAL CUBICS IN THE PROJECTIVE PLANE

We provide here an explicit example of the nodal degeneration formula and the relative splitting formula to compute a nodal Gromov-Witten invariant.

Let $X = \mathbb{P}^2$ be the complex projective plane, and let $\Gamma$ be the $X$-valued graph consisting of a single vertex $v$ with a loop $e$ and 8 legs, as illustrated in Figure A.1. Furthermore, let $g(v) = 0$ and $\beta(v) = 3 \in H_2^+(\mathbb{P}^2) = \mathbb{Z}_{\ge 0}$. We first directly compute the nodal Gromov-Witten invariant

$$ \left\langle \prod_{i=1}^{8} \tau_{0}(p) \right\rangle_{\Gamma}^{\mathbb{P}^{2}}, \qquad (A.1) $$

where $p \in H^4(\mathbb{P}^2)$ is the class of a point. Afterwards, we apply the nodal degeneration and the relative splitting formula to find the same result.
<figure><img src="image_25.png" /><figcaption>FIGURE A.1. The graph Γ on the left and Γ \ e on the right.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE A.2. A cubic in $\mathbb{P}^2$ degenerating to the union of a conic in $\mathbb{P}^2$ and a curve of class $D_0 + 3F$ in $\mathbb{F}_1$.</figcaption></figure>

The Künneth decomposition of the class of the diagonal $\Delta \subset \mathbb{P}^2 \times \mathbb{P}^2$ is

$$[\Delta] = 1 \otimes p + H \otimes H + p \otimes 1,$$

where $H \in H^2(\mathbb{P}^2)$ is the class of a line. Hence, by the splitting formula (3.5), we obtain

$$ \left\langle \prod_{i=1}^{8} \tau_0(p) \right\rangle_{\Gamma}^{\mathbb{P}^2} = \left\langle (\tau_0(1)\tau_0(p) + \tau_0(H)\tau_0(H) + \tau_0(p)\tau_0(1)) \prod_{i=1}^{8} \tau_0(p) \right\rangle_{\Gamma \setminus e}^{\mathbb{P}^2}. $$

As the graph $\Gamma \setminus e$ has no edges, the right side of the above splitting formula consists of 10-pointed, genus 0, degree 3 Gromov-Witten invariants of $\mathbb{P}^2$. As there are no rational cubic passing through 9 points in general position in $\mathbb{P}^2$, the contribution from the terms involving $\tau_0(1)\tau_0(p)$ is zero. After applying the divisor equation to remove the two insertions of $\tau_0(H)$ and using the general fact that there are 12 rational cubics passing through 8 points in general position in $\mathbb{P}^2$, we obtain

$$ \left\langle \prod_{i=1}^{8} \tau_0(p) \right\rangle_{\Gamma}^{\mathbb{P}^2} = 3 \times 3 \times 12 = 108. $$

We consider next the degeneration of $\mathbb{P}^2$ to the normal cone of a line $L$, so that the special fiber is given by the union of $\mathbb{P}^2$ and the Hirzebruch surface

$$ \mathbb{F}_1 := \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(1)) $$
<figure><img src="image_27.png" /><figcaption>FIGURE A.3. All splittings $\sigma = (\gamma_1, \gamma_2)$ contributing to the nodal degeneration formula. The graphs above the dotted lines are the $(\mathbb{F}_1, D_0)$-valued stable graphs denoted by $\gamma_1$, and below are the $(\mathbb{P}^2, L)$-valued stable graphs denoted by $\gamma_2$.</figcaption></figure>

formed by identifying $L \subset \mathbb{P}^2$ with the section $D_0$ of $\mathbb{F}_1$ with self-intersection $-1$. We distribute 4 of the 8 point constraints on the general fiber to the $\mathbb{P}^2$-component, while the other 4 point constraints are sent to the $\mathbb{F}_1$-component of the special fiber.

In the chosen degeneration, the curves in the special fiber which contribute to the nodal degeneration formula can only decompose as a (possibly singular) conic in the $\mathbb{P}^2$-component and a curve of class $D_0 + 3F$ in $\mathbb{F}_1$, as illustrated in Figure A.2. Indeed, on the $\mathbb{P}^2$ component, we have 4 point constraints, and there is no line through 4 general points. Moreover, if we had a cubic in $\mathbb{P}^2$, then on the $\mathbb{F}_1$ component, we would have a curve whose class is $3F$. However, there is no curve of class $3F$ passing through 4 general points in $\mathbb{F}_1$. Hence, the curves in the $\mathbb{P}^2$-component of the special fiber contributing to the nodal degeneration formula are necessarily conics. Then, the curves in the $\mathbb{F}_1$-component must be of class $D_0 + 3F$, since $(D_0 + 3F) \cdot D_0 = 2$ and $(D_0 + 3F) \cdot F = 1$.

We display in Figure A.3 all possible splittings $\sigma = (\gamma_1, \gamma_2)$ of $\Gamma$ contributing to the nodal degeneration formula. The contributions $N_\sigma$ associated to each $\sigma$ is computed below using (2.13) of Theorem 2.3:

(i) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(i) with $\beta(v_1) = D_0 + 3F$ and $\beta(v_2) = \beta(v'_2) = 1$. We have

$$N_{\sigma} = 1 \times 2 \times 3 \times 1 = 6,$$
where the first factor is $m(\sigma) = 1$, the factor 2 comes from the ordering of the half-edges of $e$, the factor 3 is the number of (unordered) pairs of lines in the pencil of conics passing through 4 general points in $\mathbb{P}^2$, and the last factor is the number of curves of class $D_0 + 3F$ passing through 6 general points (the 4 points in the interior of $\mathbb{F}_1$ and the 2 boundary points on $D_0$ which are fixed by choosing a pair of lines on the side of $\mathbb{P}^2$).

(ii) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(ii) with $\beta(v_1) = D_0 + 2F$, $\beta(v'_1) = F$, and $\beta(v_2) = 2$. We have

$$N_{\sigma} = 1 \times 2 \times 5 \times 1 = 10,$$

where the first factor is $m(\sigma) = 1$, the factor 2 comes from the ordering of the half-edges of $e$, the factor 5 is the number of reducible curves with two components in the pencil of curves of class $D_0 + 3F$ passing through 5 given points in $\mathbb{F}_1$, and the last factor 1 is the number of conics passing through 5 given points in $\mathbb{P}^2$. The factor 5 can be computed as follows: the total space of the pencil of curves of class $D_0+3F$ and passing through 5 points in $\mathbb{F}_1$ is the blow-up $\text{Bl}_5(\mathbb{F}_1)$ of $\mathbb{F}_1$ in 5 points, which has Euler characteristic 9. On the other hand, a $\mathbb{P}^1$-fibration over $\mathbb{P}^1$ with $k$ reducible fibers with two components has Euler characteristic

$$(2 - k) \times 2 + k \times 3 = k + 4.$$

The equation $k + 4 = 9$ implies $k = 5$.

(iii) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(iii) with $\beta(v_1) = D_0 + 3F$ and $\beta(v_2) = 2$. We have

$$N_{\sigma} = 2 \times 4 \times 2 \times 1 = 16,$$

where the first factor is $m(\sigma) = 2$, the factor 4 comes from the relative splitting formula of Theorem 3.10 to compute the nodal invariant of type $\gamma_2$ in $\mathbb{P}^2$ (the factor 4 comes from the divisor equation applied to remove the only contributing term $H \otimes H$ of the diagonal of $\mathbb{P}^2$ in the relative splitting formula$^{15}$), the factor 2 is the number of conics through 4 points in $\mathbb{P}^2$ tangent to a given line, and the last factor 1 is the number of curves of class $D_0 + 3F$ passing through 4 points in $\mathbb{F}_1$ and tangent to $D_0$ at a fixed point.

<footnote>$^{15}$The correction term is zero as the contact point with $D_0$ is already fixed, whereas the insertion of the diagonal $\Delta_{D_0} = 1 \otimes p + p \otimes 1$ would also impose a constraint on the contact point.</footnote>
(iv) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(iv) with $\beta(v_1) = D_0 + 3F$ and $\beta(v_2) = 2$. We have

$$N_{\sigma} = 2 \times 2 \times 5 \times 1 = 20,$$

where the first factor is $m(\sigma) = 2$, the second factor 2 is the number of conics through 4 points in $\mathbb{P}^2$ tangent to a given line, the factor 5 comes from the relative splitting formula of Theorem 3.10 to compute the nodal invariant of type $\gamma_1$ in $\mathbb{F}_1$ (the factor 5 comes from the divisor equation<fnref n="16" /> applied to remove the only contributing terms $F \otimes (D_0 + F) + D_0 \otimes F$ of the diagonal of $\mathbb{F}_1$). The last factor 1 is the number of curves of class $D_0 + 3F$ passing through 4 general points and tangent to $D_0$ at a fixed point.

(v) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(v) with $\beta(v_1) = D_0 + 3F$ and $\beta(v_2) = \beta(v'_2) = 1$, and let $\sigma'$ be the same splitting except that the loop is on the vertex $v'_2$. We have

$$N_{\sigma} + N_{\sigma'} = 1 \times 3 \times (1+1) \times 1 = 6,$$

where the first factor is $m(\sigma) = 1$, the factor 3 is the number of (unordered) pairs of lines in the pencil of conics passing through 4 points in $\mathbb{P}^2$, the factor $(1+1)$ comes from the relative splitting formula applied to the loop $e$ adjacent to either $v_2$ or $v'_2$ (the factor 1 comes from the divisor equation applied to remove the only contributing term $H \otimes H$ of the diagonal of $\mathbb{P}^2$), and the last factor 1 is the number of curves of class $D_0 + 3F$ passing through 6 given points.

(vi) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(vi) with $\beta(v_1) = D_0 + 3F$, and $\beta(v_2) = \beta(v'_2) = 1$. We have

$$N_{\sigma} = 1 \times 3 \times 5 \times 1 = 15,$$

obtained by proceeding analogously as in (v), except that now the loop is on $v_2$. So, the term $F \otimes (D_0 + F) + D_0 \otimes F$ of the diagonal of $\mathbb{F}_1$ contributes to the divisor equation (as in (iv)).

(vii) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(vii) with $\beta(v_1) = D_0 + 2F$, $\beta(v'_1) = F$, and $\beta(v_2) = 2$ as in (ii), and let $\sigma'$ be the same splitting except that the loop is on the vertex $v_1$. We have

$$N_{\sigma} + N_{\sigma'} = 1 \times 5 \times (3+0) \times 1 = 15,$$

where the first factor is $m(\sigma) = 1$, the factor 5 is the number of reducible curves with two components in the pencil of curves of class $D_0+3F$ passing

<footnote n="16">¹⁶We have $(D_0 + 3F) \cdot F = 1$, $(D_0 + 3F) \cdot (D_0 + F) = 3$, and $(D_0 + 3F) \cdot D_0 = 2$.</footnote>
through 5 given points in $\mathbb{F}_1$ (as in (ii)), the factor $(3+0)$ comes from the diagonal of $\mathbb{F}_1$ in the divisor equation, with either 3 or 0 depending on $e$ adjacent to $v_1$ or $v'_1$, and the last factor 1 is the number of conics through 5 points in $\mathbb{P}^2$.

(viii) Let $\sigma = (\gamma_1, \gamma_2)$ be the splitting of $\Gamma$ as illustrated in Figure A.3(vii) with $\beta(v_1) = D_0 + 2F$, $\beta(v'_1) = F$ and $\beta(v_2) = 2$. We have

$$N_{\sigma} = 1 \times 5 \times 4 \times 1 = 20,$$

computed as in (vii), except that the loop is on $v_2$, and so it is the diagonal of $\mathbb{P}^2$ which contributes to the divisor equation.

After summing the above 8 contributions, we obtain

$$6 + 10 + 16 + 20 + 6 + 15 + 15 + 20 = 108,$$

as expected.

The above example clearly illustrates why it is essential for the nodal degeneration formula and the relative splitting formula to work with nodal relative invariants defined using the moduli stack $\mathcal{P}_{\gamma}(X, D)$, where the imposed nodes are not permitted to lie in the singular locus of the expanded target, and not with the full moduli stack $\overline{\mathcal{M}}_{\gamma}(X, D)$.

For instance, in case (iii), we have on the $\mathbb{P}^2$-component the moduli stack of genus 1 degree 2 relative stable maps to $(\mathbb{P}^2, L)$ tangent to $L$ and passing through 4 given general points. A general element of this moduli space consists of one of the two conics $C$ in $\mathbb{P}^2$ tangent to $L$ and passing through the 4 given points, to which is attached a genus 1 component contracted to a point $p \in C$. This is a legitimate relative stable map as long as $p$ is away from the tangent point of $C$ with $L$, as illustrated on the left of Figure A.4. When $p$ goes to the tangent point, the target expands, and the genus 1 curve contracts to a degree 2 cover of the $\mathbb{P}^1$-fiber of the bubble over the tangent point, fully ramified along the two sections of the bubble. Another possibility is that the genus 1 curve itself maps as a degree 2 cover of the $\mathbb{P}^1$-fiber, as illustrated in the middle of Figure A.4. There is a 1-parameter family of such degree 2 covers (parameterized by the relative position on $\mathbb{P}^1$ of the two extra ramification points of the cover). When the ramification point goes to a section of the bubble, and a second bubble appears, creating the curve illustrated on the right of Figure A.4. In particular, this curve has 2 non-separating nodes mapping to the singular locus of the expanded target.

Hence, $\overline{\mathcal{M}}_{\gamma_2}(\mathbb{P}^2, L)$ is the disjoint union of $\mathcal{P}_{\gamma_2}(\mathbb{P}^2, L)$ with a set of isolated maps corresponding to geometry of the rightmost configuration in Figure A.4. A
<figure><img src="image_28.png" /><figcaption>FIGURE A.4. Degree 2 genus 1 relative stable maps to ($\mathbb{P}^2$, L)</figcaption></figure>

similar issue arises in case (iv). Including these prohibited maps would produce
an error of 16 in the nodal degeneration formula.

It is interesting to note that if we interpret (A.1) as a genus 1 Gromov-Witten invariant with insertion of the class $2\delta_0 = 24\lambda_1$ and apply the ordinary degeneration formula with insertion of $24\lambda_1$, as done in [11] more generally with insertion of $(-1)^g\lambda_g$, then these extra curves do contribute 16. However, with the insertion of $\lambda_1$, the cases (i) and (ii) contributing 16 in our discussion do not contribute because $\lambda_1$ vanishes on families of curves containing a cycle, and so we obtain the same total answer, as expected.

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UNIVERSITY OF GEORGIA, DEPARTMENT OF MATHEMATICS, ATHENS, GA 30605
*Email address:* Hulya.Arguz@uga.edu

UNIVERSITY OF GEORGIA, DEPARTMENT OF MATHEMATICS, ATHENS, GA 30605
*Email address:* Pierrick.Bousseau@uga.edu

DEPARTEMENT MATHEMATIK, ETH ZÜRICH, 8092 ZÜRICH, SWITZERLAND
*Email address:* rahul@math.ethz.ch

UNIVERSITÉ DE VERSAILLES SAINT-QUENTIN-EN-YVELINES, VERSAILLES, 78000, FRANCE
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