3.1 Hilbert schemes of points and Heisenberg algebra actions

Let X be a smooth projective complex surface, and let $ X^{[n]}$ be the Hilbert scheme of points in X. An element in $ X^{[n]}$ is represented by a length- $n \ 0$ -dimensional closed subscheme $\xi $ of X. For $\xi \in X^{[n]}$ , let $I_{\xi }$ and $\mathcal O_{\xi }$ be the corresponding sheaf of ideals and structure sheaf, respectively. It is known [Reference Fogarty10, Reference Iarrobino16] that $ X^{[n]}$ is a smooth irreducible variety of dimension $2n$ . The boundary of $ X^{[n]}$ is defined to be

$$ \begin{align*}B_n = \left \{ \xi \in X^{[n]}|\, |\mathrm{Supp}{(\xi)}| < n \right \}. \end{align*} $$

For fixed distinct points $x_1, \ldots , x_{n-1} \in X$ , define the curve

(3.1) $$ \begin{align} \beta_n = \left \{ \xi + x_2 + \ldots + x_{n-1} \in X^{[n]} | \mathrm{Supp}(\xi) = \{x_1\} \right \} \cong {\mathbb P}^1. \end{align} $$

We also regard $\beta _n$ as a homology class in $H_2( X^{[n]}, \mathbb Z )$ . For a subset $Y \subset X$ , define

$$ \begin{align*}M_n(Y) = \{ \xi \in X^{[n]}| \, \mathrm{Supp}(\xi) \text{ is a point in } Y\}. \end{align*} $$

Sending an element in $ X^{[n]}$ to its support (counting with multiplicities) in the n-th symmetric product $X^{(n)}$ of X, we obtain the Hilbert-Chow morphism

$$ \begin{align*}\rho_n: X^{[n]} \rightarrow X^{(n)} \end{align*} $$

which is a crepant resolution of singularities. A curve in $ X^{[n]}$ is contracted by $\rho _n$ if and only if it is homologous to $d\beta _n$ for some positive integer d.

Grojnowski [Reference Grojnowski13] and Nakajima [Reference Nakajima34] geometrically constructed a Heisenberg algebra action on the cohomology of the Hilbert schemes $ X^{[n]}$ . Denote the Heisenberg operators by $ {\mathfrak a}_m(\alpha )$ where $m \in \mathbb Z $ and $\alpha \in H^*(X, {\mathbb C} )$ . Put

$$ \begin{align*}{\mathbb H}_X = \bigoplus_{n=0}^{+ \infty} H^*( X^{[n]}, {\mathbb C} ). \end{align*} $$

The operators ${\mathfrak a}_m(\alpha ) \in \mathrm {End}(\mathbb H_X)$ satisfy the following commutation relation:

(3.2) $$ \begin{align} [{\mathfrak a}_m(\alpha), {\mathfrak a}_n(\beta)] = -m \cdot \delta_{m,-n} \cdot \langle \alpha, \beta \rangle \cdot \mathrm{Id}_{\mathbb H_X} \end{align} $$

where we have used $\delta _{m,-n}$ to denote $1$ if $m=-n$ and $0$ otherwise. The space ${\mathbb H}_X$ is an irreducible representation of the Heisenberg algebra generated by the operators $ {\mathfrak a}_m(\alpha )$ with the highest weight vector being $|0\rangle = 1 \in H^*(X^{[0]}, {\mathbb C} ) = {\mathbb C} $ . In particular, $H^*( X^{[n]}, {\mathbb C} )$ is the linear span of Heisenberg monomial classes

(3.3) $$ \begin{align} {\mathfrak a}_{-n_1}(\alpha_1) \cdots {\mathfrak a}_{-n_k}(\alpha_k) |0\rangle \end{align} $$

where $k \ge 0, n_1, \ldots , n_k> 0$ and $\alpha _1, \ldots , \alpha _k \in H^*(X, {\mathbb C} )$ .

Fix closed real cycles $X_1, \ldots , X_k$ of the surface X in general position in the sense that any subset of the $X_i$ ’s meet transversally in the expected dimension. Define

$$ \begin{align*} W(n_1, X_1; \ldots; n_k, X_k) \subset X^{[n]} \end{align*} $$

to be the closed subset consisting of all $\xi \in X^{[n]}$ which admit filtrations

$$ \begin{align*} \xi = \xi_k \supset \ldots \supset \xi_1 \supset \xi_0 = \emptyset \end{align*} $$

with $\ell (\xi _i) = \ell (\xi _{i-1}) + n_i$ and

(3.4) $$ \begin{align} \mathrm{Supp}(I_{\xi_{i-1}}/I_{\xi_{i}}) = x_i \in X_i \end{align} $$

for $1 \le i \le k$ . Let $ W(n_1, X_1; \ldots; n_k, X_k)^0 \subset W(n_1, X_1; \ldots; n_k, X_k)$ be the open subset consisting of all $\xi \in W(n_1, X_1; \ldots; n_k, X_k)$ such that the points $x_1, \ldots , x_k$ in (3.4) are distinct.

Lemma 3.1 [Reference Qin38, Proposition 3.16].

Let $\ell , k \ge 0$ , $s_i \ge 0$ ( $1 \le i \le \ell $ ), $n_i> 0$ ( $1 \le i \le k$ ). Let $\alpha _1, \ldots , \alpha _k \in \displaystyle {\bigoplus _{i=1}^4 H^i(X, {\mathbb C} )}$ be represented by the cycles $X_1, \ldots , X_k \subset X$ , respectively, such that $X_1, \ldots , X_k$ are in general position. Then,

$$ \begin{align*} \displaystyle{\left ( \prod_{i=1}^{\ell} \frac{{\mathfrak a}_{-i}(1_X)^{s_i}}{s_i!} \right ) \left ( \prod_{i=1}^k {\mathfrak a}_{-n_i}(\alpha_i) \right ) |0\rangle} \end{align*} $$

is represented by the closure of

(3.5) $$ \begin{align} W(\underbrace{1, X; \ldots; 1, X}_{s_1 \,\,\, \mathrm{times}}; \ldots; \underbrace{\ell, X; \ldots; \ell, X}_{s_{\ell} \,\,\, \mathrm{times}}; n_1, X_1; \ldots; n_k, X_k)^0. \end{align} $$

It follows that $1_{ X^{[n]}} = 1/n! \cdot {\mathfrak a}_{-1}(1_X)^n|0\rangle \in H^0( X^{[n]}, {\mathbb C} )$ , where $1_X$ denotes the fundamental cohomology class of the surface X, and

(3.6) $$ \begin{align} \beta_n &= {\mathfrak a}_{-2}(x) {\mathfrak a}_{-1}(x)^{n-2} |0\rangle, \end{align} $$

(3.7) $$ \begin{align} B_n &= \frac{1}{(n-2)!} {\mathfrak a}_{-1}(1_X)^{n-2} {\mathfrak a}_{-2}(1_X) |0\rangle, \end{align} $$

where x denotes the fundamental cohomology class of a point $x \in X$ . For simplicity, we do not distinguish a homology class and its Poincaré dual.

Let $\tau _2: X \to X^2$ be the diagonal embedding and $\tau _{2*}: H^*(X, {\mathbb C} ) \to H^*(X^2, {\mathbb C} )$ be the induced map. For $\alpha \in H^*(X, {\mathbb C} )$ and $m_1, m_2 \in \mathbb Z $ , define

$$ \begin{align*}{\mathfrak a}_{m_1} {\mathfrak a}_{m_2} (\tau_{2*}\alpha) = \sum_i {\mathfrak a}_{m_1}(\alpha_{i,1}) {\mathfrak a}_{m_2}(\alpha_{i,2}) \end{align*} $$

if $\tau _{2*}\alpha = \sum _i \alpha _{i,1} \otimes \alpha _{i,2}$ under the Künneth decomposition of $H^*(X^2, {\mathbb C} )$ . For $n \in \mathbb Z $ and $\alpha \in H^*(X, {\mathbb C} )$ , define the linear operator $\mathfrak L_n(\alpha ) \in \mathrm {End}(\mathbb H_X)$ by

(3.8) $$ \begin{align} &\mathfrak L_n = \left\{ {\displaystyle} \begin{array}{ll} -\dfrac{1}{2} \cdot \displaystyle{\sum_{m \in \mathbb Z }} {\mathfrak a}_m {\mathfrak a}_{n-m} \tau_{2*}, & \text{ if } n \ne 0, \\ \displaystyle{-\sum_{m> 0}} {\mathfrak a}_{-m} {\mathfrak a}_m \tau_{2*}, &\text{ if } n = 0. \end{array} \right. \end{align} $$

We have the commutation relation

(3.9) $$ \begin{align} [\mathfrak L_m(\alpha), {\mathfrak a}_n(\beta)] = -n \cdot {\mathfrak a}_{m+n}(\alpha \beta). \end{align} $$

Lehn [Reference Lehn21] defined the boundary operator $ \mathfrak d \in \mathrm {End}(\mathbb H_X)$ by putting

(3.10) $$ \begin{align} \mathfrak d \cdot \gamma_n = -\frac{1}{2} B_n \cup \gamma_n \end{align} $$

for $\gamma _n \in H^*( X^{[n]}, {\mathbb C} )$ . For a linear operator $\mathfrak f \in \mathrm {End}(\mathbb H_X)$ , define its derivative $\mathfrak f'$ by

$$ \begin{align*}\mathfrak f' \,\, = \,\, [\mathfrak d, \mathfrak f]. \end{align*} $$

A fundamental result proved in [Reference Lehn21] states that

(3.11) $$ \begin{align} {\mathfrak a}_n'(\alpha) = n \cdot \mathfrak L_n(\alpha) - \frac{n(|n|-1)}{2} \cdot {\mathfrak a}_n(K_X \alpha). \end{align} $$

3.2 $1$ -point and $2$ -point genus- $0$ extremal Gromov-Witten invariants

In this subsection, let X be a simply connected smooth projective surface. We start with the $1$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ .

Lemma 3.2 [Reference Li and Qin27, Theorem 3.5].

Let X be a simply connected smooth projective surface. Let $n \ge 2$ , $d\ge 1$ and $\gamma \in H^*( X^{[n]}, {\mathbb C} )$ be a Heisenberg monomial class (3.3). Then, $\langle \gamma \rangle _{0, d\beta _n}=0$ unless $\gamma = {\mathfrak a}_{-2}(\alpha ){\mathfrak a}_{-1}(x)^{n-2}|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ . Moreover, if $\gamma = {\mathfrak a}_{-2}(\alpha ){\mathfrak a}_{-1}(x)^{n-2}|0\rangle $ for some $\alpha \in H^2(X, {\mathbb C} )$ , then

$$ \begin{align*}\langle \gamma\rangle_{0, d\beta_n} = \frac{2}{d^2} \cdot \langle K_X, \alpha \rangle. \end{align*} $$

The $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ have been studied in [Reference Li and Li23] via cosection localizations [Reference Kiem and Li17]. By abusing notations, denote

$$ \begin{align*}[\varphi: (C; p_1, \ldots, p_k) \to X^{[n]}] \in \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta_n) \end{align*} $$

by $\varphi $ . Since $\varphi _*[C] = d \beta _n$ , the composition $\rho _n \circ \varphi $ is a constant map. Let

$$ \begin{align*}\mathrm{Spt}: \overline {\mathfrak M}_{g, k}( X^{[n]}, d \beta_n) \to X^{(n)} \end{align*} $$

be the induced map. If $\mathrm {Spt}(\varphi ) = \sum _{i=1}^{\ell } n_i x_i \in X^{(n)}$ where $x_1, \ldots , x_{\ell }$ are distinct, then the morphism $\varphi $ factors through the product of punctual Hilbert schemes:

(3.12) $$ \begin{align} \varphi = (\varphi_1, \ldots, \varphi_{\ell}): C \to \prod_{i=1}^{\ell} M_{n_i}(x_i) \subset X^{[n]} \end{align} $$

where $\varphi _i$ is a morphism from C to $M_{n_i}(x_i)$ . The collection

(3.13) $$ \begin{align} \varphi = (\varphi_1, \ldots, \varphi_{\ell}) \end{align} $$

is defined to be the standard decomposition of $\varphi $ , and the point $x_i$ is called the support of $\varphi _i$ . Note that the collection $\{\varphi _1, \ldots , \varphi _{\ell } \}$ is unique up to the ordering of the $\varphi _i$ ’s. Fix a meromorphic section $\theta $ of $\mathcal O_X(K_X)$ . Let $D_0$ and $D_{\infty }$ be the vanishing and pole divisors of $\theta $ , respectively.

Let $(\mu ^1, \mu ^2, \mu ^3)$ denote a triple of partitions with $|\mu ^1| + |\mu ^2| + |\mu ^3| = n$ . Let $r = \ell (\mu ^1)$ , $s = \ell (\mu ^2)$ and $t = \ell (\mu ^3)$ be the lengths. For cohomology classes $\mathbf {c}_1, \ldots , \mathbf {c}_s \in H^2(X, {\mathbb C} )$ , define the class $A_{\mathbf {c}}^{\mu } \in H^*( X^{[n]}, {\mathbb C} )$ by

(3.14) $$ \begin{align} A_{\mathbf{c}}^{\mu} = \prod_{i=1}^r {\mathfrak a}_{-\mu_i^1}(x) \cdot \prod_{j=1}^s {\mathfrak a}_{-\mu_j^2}(\mathbf{c}_j) \cdot \prod_{k=1}^t {\mathfrak a}_{-\mu_k^3}(1_X) |0\rangle. \end{align} $$

For a part $\mu _j^2$ of $\mu ^2$ , let $A_{\mathbf {c}}^{\mu - \mu _j^2}$ be the cohomology class in $H^*(X^{[n - \mu _j^2]}, {\mathbb C} )$ obtained from $A_{\mathbf {c}}^{\mu }$ with the factor ${\mathfrak a}_{-\mu _j^2}(\mathbf {c}_j)$ deleted. We similarly define $A_{\mathbf {c}}^{\mu - \mu _i^3}$ .

The following lemma summarizes some of the main results in [Reference Li and Li23] and computes the $2$ -point genus- $0$ extremal Gromov-Witten invariants of $ X^{[n]}$ .

Lemma 3.4 [Reference Li and Li23].

Let $d \ge 1$ . Assume that $\big \langle A_{\mathbf {e}}^{\lambda }, A_{\mathbf {c}}^{\mu } \big \rangle _{0, d \beta _n} \ne 0$ . Then,

$$ \begin{align*} \ell(\lambda^3) &= \ell(\mu^1) + \delta \\ \ell(\mu^3) &= \ell(\lambda^1) + (1-\delta) \end{align*} $$

where $\delta =0$ or $1$ . If $\delta =0$ , then $\lambda ^3 = \mu ^1$ , and there exists an integer $\ell = \mu _i^3 = \lambda _j^2$ for some i and j such that the partition $\lambda ^1$ is obtained from $\mu ^3$ with $\ell $ deleted, and the partition $\mu ^2$ is obtained from $\lambda ^2$ with $\ell $ deleted; moreover,

(3.15) $$ \begin{align} \big \langle A_{\mathbf{e}}^{\lambda}, A_{\mathbf{c}}^{\mu} \big \rangle_{0, d \beta_n} = \sum_{\ell = \mu_i^3 = \lambda_j^2} \big \langle A_{\mathbf{e}}^{\lambda - \lambda_j^2}, A_{\mathbf{c}}^{\mu - \mu_i^3} \big \rangle \cdot \langle K_X, \mathbf{e}_j \rangle \cdot c_{\ell, d} \end{align} $$

where the universal constant $c_{\ell , d}$ is defined by the equation

(3.16) $$ \begin{align} \sum_{d \ge 1} d \, c_{\ell, d} \, q^d = (-1)^{\ell} \ell^2 \left ( \frac{\ell (-q)^{\ell}}{(-q)^{\ell}-1} - \frac{q}{1+q} \right ). \end{align} $$

5.1 The contracted $( {\mathbb C} ^*)^2$ -invariant curves in $ X^{[3]}$ for a toric surface X

Let X be a smooth projective toric surface. In this subsection, we will write down all the invariant curves contracted by the Hilbert-Chow morphism $\rho _3: X^{[3]} \to X^{(3)}$ .

We begin with some standard setups. The surface X is determined by a fan $\Sigma $ which is a finite collection of strongly convex rational polyhedral cones $\sigma $ contained in $N = \mathrm {Hom}(M, \mathbb Z)$ , where $M \cong \mathbb Z^2$ . So X is obtained by gluing together affine toric varieties $X_{\sigma }$ and $X_{\tau }$ along $X_{\sigma \cap \tau }$ for $\sigma , \tau \in \Sigma $ . The coordinate ring of $X_{\sigma }$ is $ {\mathbb C} [\sigma ^{\vee } \cap M]$ , which is the $ {\mathbb C} $ -algebra with generators $\chi ^m$ for $m \in \sigma ^{\vee } \cap M$ and multiplication defined by $\chi ^m \cdot \chi ^{m'} = \chi ^{m+m'}$ . By definition, $\sigma ^{\vee } \cap M$ is the set of elements $m \in M$ satisfying $v(m) \ge 0$ for all $v \in \sigma $ . The torus

$$ \begin{align*}{\mathbb T} = ( {\mathbb C} ^*)^2 \end{align*} $$

acts on X with finitely many fixed points $x_1, \ldots , x_{\chi (X)}$ . For each i, the point $x_i$ lies in $U_i := X_{\sigma _i}$ for some $\sigma _i \in \Sigma $ . As X is smooth and $U_i$ possesses a unique fixed point $x_i$ , $U_i$ is isomorphic to the affine plane with $x_i$ corresponding to the origin. Let $u_i, v_i$ be the affine coordinates of $U_i$ . Assume that

$$ \begin{align*}(s,t)(u_i, v_i) = (\lambda_i(s,t)u_i, \mu_i(s,t)v_i) \end{align*} $$

for $(s, t) \in {\mathbb T}$ , where $\lambda _i(s,t)$ and $\mu _i(s,t)$ are two independent characters of $ {\mathbb T}$ . Denote the weights of $\lambda _i(s,t)$ and $\mu _i(s,t)$ by $w_i$ and $z_i$ , respectively, that is,

$$ \begin{align*}w_i = c_1 \big (\lambda_i(s,t) \big ), \quad z_i = c_1 \big (\mu_i(s,t) \big ) \end{align*} $$

in the equivariant Chow group $A^{ {\mathbb T}}_*(pt)$ . By the Atiyah-Bott localization formula,

(5.1) $$ \begin{align} K_X^2 = \int_X c_1(T_X)^2 = \sum_{i=1}^{\chi(X)} \frac{(w_i + z_i)^2}{w_iz_i} \end{align} $$

noting that $T_{x_i, X} = \big ( \lambda _i(s,t) \big )^{-1} + \big ( \mu _i(s,t) \big )^{-1}$ as representations.

The $ {\mathbb T}$ -action on the toric surface X induces a $ {\mathbb T}$ -action on the Hilbert scheme $ X^{[3]}$ with a finite number of fixed points. The $ {\mathbb T}$ -fixed points in $ X^{[3]}$ are enumerated as follows. For each $1 \le i \le \chi (X)$ , there are three $ {\mathbb T}$ -fixed points

$$ \begin{align*}Q_{i, 0}, \qquad Q_{i, 1}, \qquad Q_{i, 2} \end{align*} $$

in $M_3(x_i) \subset X^{[3]}$ corresponding, respectively, to the partitions $(2,1)$ , $(3)$ and $(1,1,1)$ of $3$ . The corresponding ideals are $(v_i^2, v_iu_i, u_i^2)$ , $(v_i^3, u_i)$ and $(v_i, u_i^3)$ . Also for each ordered pair $(i, j)$ with $i, j \in \{1, \ldots , \chi (X)\}$ and $i \ne j$ , we have two fixed points

$$ \begin{align*}R_{i,j}^{(1)} = \xi_{i, 1} + x_j, \qquad R_{i,j}^{(2)} = \xi_{i, 2} + x_j \end{align*} $$

in $ X^{[3]}$ , where $\xi _{i, 1}, \xi _{i, 2} \in M_2(x_i)$ correspond to the ideals $(v_i^2, u_i), (v_i, u_i^2)$ , respectively. Furthermore, whenever $i, j, k \in \{1, \ldots , \chi (X)\}$ are mutually distinct, $x_i+x_j+x_k \in X^{[3]}$ is a $ {\mathbb T}$ -fixed point in $ X^{[3]}$ . Denote the tangent space of $ X^{[3]}$ at $\xi \in X^{[3]}$ by $T_{\xi }$ . As representations of $ {\mathbb T}$ , we have the decompositions (see [Reference Ellingsrud and Strømme9]):

(5.2) $$ \begin{align} T_{Q_{i, 0}} &=2\lambda_i^{-1} + 2\mu_i^{-1} + \lambda_i^{-2}\mu_i + \lambda_i \mu_i^{-2}, \end{align} $$

(5.3) $$ \begin{align} T_{Q_{i, 1}} &= \lambda_i^{-1}\mu_i^2 + \lambda_i^{-1}\mu_i+ \lambda_i^{-1} + \mu_i^{-3} + \mu_i^{-2} + \mu_i^{-1}, \end{align} $$

(5.4) $$ \begin{align} T_{Q_{i, 2}} &=\lambda_i^{-3} + \lambda_i^{-2} + \lambda_i^{-1} + \lambda_i^2\mu_i^{-1} + \lambda_i \mu_i^{-1} + \mu_i^{-1}, \end{align} $$

(5.5) $$ \begin{align} T_{R_{i,j}^{(1)}} &= \lambda_i^{-1}\mu_i + \lambda_i^{-1} + \mu_i^{-2} + \mu_i^{-1} + \lambda_j^{-1}+ \mu_j^{-1}, \end{align} $$

(5.6) $$ \begin{align} T_{R_{i,j}^{(2)}} &= \lambda_i^{-2} + \lambda_i^{-1}+ \lambda_i\mu_i^{-1} + \mu_i^{-1} + \lambda_j^{-1} + \mu_j^{-1}. \end{align} $$

There are exactly three $ {\mathbb T}$ -invariant curves $C_{0, 1}^{(i)}$ , $C_{0, 2}^{(i)}$ and $C_{1, 2}^{(i)}$ in $M_3(x_i)$ . Namely, $C_{0, 1}^{(i)}$ goes through $Q_{i, 0}$ and $Q_{i, 1}$ , and is the fixed locus of $\ker (\lambda _i \mu _i^{-2})$ ; $C_{0, 2}^{(i)}$ goes through $Q_{i, 0}$ and $Q_{i, 2}$ , and is the fixed locus of $\ker (\lambda _i^{-2}\mu _i)$ ; $C_{1, 2}^{(i)}$ goes through $Q_{i, 1}$ and $Q_{i, 2}$ , and is the fixed locus of $\ker (\lambda _i^{-1}\mu _i)$ . The following is from [Reference Edidin, Li and Qin8].

Next, let $f: {\mathbb P}^1 \to X^{[3]}$ be a degree-d morphism such that the image is one of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 and f is totally ramified at the two $ {\mathbb T}$ -fixed points in $f({\mathbb P}^1)$ . The Euler characteristic $\chi (f^*T_{ X^{[3]}})$ (as a representation) has been computed in [Reference Edidin, Li and Qin8]. When $f({\mathbb P}^1) = C_{0,1}^{(i)}$ , we have

(5.7) $$ \begin{align} \chi(f^*T_{ X^{[3]}}) &=(1 + \lambda_i^{-1}\mu_i^2 +\lambda_i\mu_i^{-2} + \lambda_i^{-1}\mu_i + \mu_i^{-1} + \lambda_i^{-1} + \mu_i^{-1} - \lambda_i^{-1}\mu_i^{-1}) \nonumber \\ &\quad + (\lambda_i^{-1}\mu_i^2 + 1 + \lambda_i^{-1}\mu_i - \lambda_i^{-2}\mu_i - \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-1})\Theta_{0,1}^{(i)} \end{align} $$

where $\Theta _{0,1}^{(i)} = \sum _{m= 1}^{d-1} (\lambda _i\mu _i^{-2})^{m/d}$ ( $\Theta _{0,1}^{(i)} = 0$ when $d=1$ ). If $f({\mathbb P}^1) = C_{0,2}^{(i)}$ , then

(5.8) $$ \begin{align} \chi(f^*T_{ X^{[3]}}) &=(1 + \mu_i^{-1}\lambda_i^2 +\mu_i\lambda_i^{-2} + \mu_i^{-1}\lambda_i + \lambda_i^{-1} + \mu_i^{-1} + \lambda_i^{-1} - \mu_i^{-1}\lambda_i^{-1}) \nonumber \\ &\quad + (\mu_i^{-1}\lambda_i^2 + 1 + \mu_i^{-1}\lambda_i - \mu_i^{-2}\lambda_i - \mu_i^{-1}\lambda_i^{-1} - \mu_i^{-1})\Theta_{0,2}^{(i)} \end{align} $$

where $\Theta _{0,2}^{(i)} = \sum _{m=1}^{d-1} (\mu _i\lambda _i^{-2})^{m/d}$ ( $\Theta _{0,2}^{(i)} = 0$ when $d = 1$ ). Let $\Theta _{1,2}^{(i)} = \sum _{m= 1}^{d-1} (\lambda _i\mu _i^{-1})^{m/d}$ with $\Theta _{1,2}^{(i)} = 0$ when $d=1$ . If $f({\mathbb P}^1) = C_{1,2}^{(i)}$ , then $\chi (f^*T_{ X^{[3]}})$ is equal to

(5.9) $$ \begin{align} &(1 + \lambda_i^{-1}\mu_i^2 + \mu_i +\lambda_i + \lambda_i^{-1}\mu_i - \lambda_i^{-2} - \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-3} - \lambda_i^{-2}\mu_i^{-1}- \lambda_i^{-1}\mu_i^{-2}) \Theta_{1,2}^{(i)} \notag\\ &\qquad\quad +\, (\lambda_i^{-1} + \mu_i^{-1}+ \lambda_i^{-1}\mu_i+1 + \lambda_i\mu_i^{-1} + \lambda_i^{-1}\mu_i^2 + \mu_i + \lambda_i + \lambda_i^2\mu_i^{-1} \notag \\ &\qquad\qquad\qquad -\, \lambda_i^{-1}\mu_i^{-1} - \lambda_i^{-2}\mu_i^{-1} -\lambda_i^{-1}\mu_i^{-2}). \end{align} $$

Finally, when $f({\mathbb P}^1) = C_{i, j}$ , then $\chi (f^*T_{ X^{[3]}})$ is equal to

(5.10) $$ \begin{align} &(1 + \lambda_i^{-1}\mu_i +\lambda_i\mu_i^{-1} + \lambda_i^{-1} + \mu_i^{-1} + \lambda_j^{-1} + \mu_j^{-1} - \lambda_i^{-1}\mu_i^{-1}) \nonumber \\ &\qquad\quad \ + (1 + \lambda_i^{-1}\mu_i - \lambda_i^{-2} - \lambda_i^{-1}\mu_i^{-1} )\Theta_{1,2}^{(i)}. \end{align} $$

5.2 $ {\mathbb T}$ -invariant stable maps, stable graphs and localizations

Let X be a smooth projective toric surface and $d \ge 1$ . For simplicity, put

$$ \begin{align*}{\overline{\mathfrak M}}_{g, r, d} = {\overline{\mathfrak M}}_{g, r}( X^{[3]}, d\beta_3). \end{align*} $$

In this subsection, using virtual localization formula, we will express the genus- $1$ extremal Gromov-Witten invariant $\langle \rangle _{1, d\beta _3}$ in terms of stable graphs.

As in [Reference Graber and Pandharipande12, Reference Kontsevich19], if $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{1, 0, d}$ is $ {\mathbb T}$ -invariant, then all the nodes, contracted components and ramification points are mapped into the $ {\mathbb T}$ -fixed point set $( X^{[3]})^{{\mathbb T}}$ . Moreover, if $\widetilde C \subset C$ is a noncontracted component, then $\widetilde C = {\mathbb P}^1$ , $f(\widetilde C)$ is one of the $ {\mathbb T}$ -invariant curves in Lemma 5.2, and $f|_{\widetilde C}$ is of the form

$$ \begin{align*}(z_0, z_1) \mapsto (z_0^{\tilde d}, z_1^{\tilde d}) \end{align*} $$

where $\tilde d = \deg (f|_{\widetilde C})$ . Therefore, to each stable map $[f: C \to X^{[3]}] \in ( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ , we can associate a stable graph $\Gamma $ as follows. The stable graph $\Gamma $ has one vertex for each connected component of $f^{-1}(( X^{[3]})^{{\mathbb T}})$ and one edge for every noncontracted component. The edge e is marked with the degree $d_e$ of f restricted to that noncontracted component $C_e = {\mathbb P}^1$ , and the connected component corresponding to a vertex v is denoted by $C_v$ . Let $V(\Gamma )$ (respectively, $E(\Gamma )$ ) denote the set of vertices (respectively, edges) of $\Gamma $ . Define the labeling map

$$ \begin{align*}\mathfrak L: \quad V(\Gamma) \longrightarrow ( X^{[3]})^{{\mathbb T}} \end{align*} $$

by putting $\mathfrak L(v) = f(C_v)$ . The vertices have an additional labeling $g(v)$ which is the arithmetic genus of $C_v$ ( $g(v) = 0$ if $C_v$ is a point) and satisfies the identity

$$ \begin{align*}1 - |V(\Gamma)| + |E(\Gamma)| + \sum_{v \in V(\Gamma)} g(v) = g = 1. \end{align*} $$

The valence of v, denoted by $\mathrm {val}(v)$ , is the number of edges connected to v. Define a flag F of the graph $\Gamma $ to be an incident edge-vertex pair $(e, v)$ . Put

$$ \begin{align*}i(F) = \mathfrak L(v). \end{align*} $$

A flag $F = (e, v)$ is defined to be stable if $2g(v) + \mathrm {val}(v) \ge 3$ . Since $\mathrm {val}(v) \ge 1$ , F is not stable if $g(v) = 0$ and $\mathrm {val}(v) = 1$ or $2$ (in these cases, the component $C_v$ is simply a point). Let $F(\Gamma )$ (respectively, $F(\Gamma )^{\mathrm {sta}}$ ) be the set of flags (respectively, stable flags) in $\Gamma $ . The edge e in $F = (e, v)$ is incident to one other vertex $v'$ . Define $j(F) = \mathfrak L(v')$ . If $\mathrm {val}(v) = 1$ , let $F(v)$ be the unique flag containing v; if $\mathrm {val}(v) = 2$ , let $F_1(v)$ and $F_2(v)$ denote the two flags containing v.

Now the connected components of $( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ are indexed by stable graphs corresponding to stable maps whose images are unions of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 and whose contracted components and special points are mapped into $( X^{[3]})^{{\mathbb T}}$ . We use $\Gamma $ to denote these stable graphs. So we have

(5.11) $$ \begin{align} ( {\overline{\mathfrak M}}_{1, 0, d})^{{\mathbb T}} = \coprod_{\Gamma} {\overline{\mathfrak M}}_{\Gamma} \end{align} $$

where ${ {\overline {\mathfrak M}}}_{\Gamma }$ denotes the connected component of $( {\overline {\mathfrak M}}_{1, 0, d})^{{\mathbb T}}$ indexed by $\Gamma $ . Let $\overline {M}_{g, n}$ be the moduli space of n-pointed genus-g stable curves. Put

$$ \begin{align*}\overline{M}_{\Gamma} = \prod\limits_{v \in V(\Gamma)} \overline{M}_{g(v), \mathrm{val}(v)} \end{align*} $$

( $\overline {M}_{0, 1}$ and $\overline {M}_{0, 2}$ are treated as points in this product). Then there is a finite map $\overline {M}_{\Gamma } \to {\overline {\mathfrak M}}_{\Gamma }$ such that $ {\overline {\mathfrak M}}_{\Gamma } = \overline {M}_{\Gamma }/\mathbf {A}_{\Gamma }$ where $\mathbf {A}_{\Gamma }$ fits in the exact sequence

(5.12) $$ \begin{align} 0 \to \prod_{e \in E(\Gamma)} {\mathbb Z}/d_e {\mathbb Z} \to \mathbf{A}_{\Gamma} \to {\mathrm{Aut}} (\Gamma) \to 0. \end{align} $$

Since a stable curve is connected, we see from the description of the $ {\mathbb T}$ -invariant curves in Lemma 5.2 that a summation over all the stable graphs $\Gamma $ breaks up as

(5.13) $$ \begin{align} \sum_{\Gamma} \,\, = \,\, \sum_{1 \leq i\neq j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \,\, +\,\, \sum_{i=1}^{\chi(X)} \,\,\sum_{\Gamma \in {\mathcal T}_{d,i}} \end{align} $$

where ${{\mathcal S}}_{d,i,j}$ is the set of all stable graphs $\Gamma $ such that $f(C) = C_{i,j}$ for every $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ , and ${\mathcal T}_{d,i}$ is the set of all stable graphs $\Gamma $ such that $f(C) \subset C_{0,1}^{(i)} \cup C_{0,2}^{(i)} \cup C_{1,2}^{(i)}$ for every $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ .

By the virtual localization formula of [Reference Graber and Pandharipande12], we have

(5.14) $$ \begin{align} \langle \rangle_{1, d\beta_3} \quad = \quad \int_{[ {\overline{\mathfrak M}}_{1, 0, d}]^{\mathrm{vir}} } 1 \quad = \quad \sum_{\Gamma} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]^{\mathrm{vir}} } \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Here, $[\overline {M}_{\Gamma }]^{\mathrm {vir}} $ is the pullback of $[ {\overline {\mathfrak M}}_{\Gamma }]^{\mathrm {vir}} $ to $M_{\Gamma }$ via the finite map $\overline {M}_{\Gamma } \to {\overline {\mathfrak M}}_{\Gamma }$ , and $e(N_{\Gamma }^{\mathrm {vir}} )$ is the pullback of the Euler class of the moving part $N_{\Gamma }^{\mathrm {vir}} $ of the tangent-obstruction complex. Let ${{\mathcal T}}^1$ and ${{\mathcal T}}^2$ be the cohomology sheaves of the restriction of the tangent-obstruction complex on $ {\overline {\mathfrak M}}_{1, 0, d}$ to $ {\overline {\mathfrak M}}_{\Gamma }$ . The fibers of ${{\mathcal T}}^1$ and ${{\mathcal T}}^2$ at a point associated to a stable map $[f: C \to X^{[3]}] \in {\overline {\mathfrak M}}_{\Gamma }$ fit into the exact sequence

$$ \begin{align*} 0 &\to \mathrm{Ext}^0(\Omega_C,{\mathcal O}_C) \to H^0(C, f^*T_{ X^{[3]}}) \to {{\mathcal T}}^1 \\ & \to \mathrm{Ext}^1(\Omega_C,{\mathcal O}_C)\to H^1(C, f^*T_{ X^{[3]}}) \to {{\mathcal T}}^2 \to 0. \end{align*} $$

To understand $H^i(C, f^*T_{ X^{[3]}})$ , consider the normalization sequence resolving the nodes of C coming from all the intersections $x_F := C_v \cap C_e$ :

$$ \begin{align*}0 \to \mathcal O_C \to \bigoplus_{v} \mathcal O_{C_v} \oplus \bigoplus_{e} \mathcal O_{C_e} \to \bigoplus_{F} \mathcal O_{x_F} \to 0. \end{align*} $$

Tensoring by $f^*T_{ X^{[3]}}$ and taking cohomology, we obtain an exact sequence

$$ \begin{align*}0 \to H^0(C, f^*T_{ X^{[3]}}) \to \bigoplus_{v} T_{\mathfrak L(v)} \oplus \bigoplus_{e} H^0(C_e, f^*T_{ X^{[3]}}) \to \bigoplus_{F} T_{i(F)} \end{align*} $$

(5.15) $$ \begin{align} \to H^1(C, f^*T_{ X^{[3]}}) \to \bigoplus_{v} H^1(C_v, f^*T_{ X^{[3]}}) \oplus \bigoplus_{e} H^1(C_e, f^*T_{ X^{[3]}}) \to 0. \end{align} $$

Note that $H^1(C_v, f^*T_{ X^{[3]}}) = H^1(C_v, \mathcal O_{C_v}) \otimes T_{\mathfrak L(v)}$ where $H^1(C_v, \mathcal O_{C_v})$ forms the dual of the Hodge bundle $\mathcal H_{g(v)}$ over $\overline {M}_{g(v), \mathrm {val}(v)}$ . By the five formulas (5.2)-(5.6), the fixed parts of $T_{i(F)}$ and $H^1(C_v, f^*T_{ X^{[3]}})$ vanish. Examining the terms in the four formulas (5.7)-(5.10) which carry negative signs, we see that the fixed part of $H^1(C_e, f^*T_{ X^{[3]}})$ also vanishes. By (5.15), the fixed part of $H^1(C, f^*T_{ X^{[3]}})$ vanishes. Thus, ${{\mathcal T}}^{2, f} = 0$ , and the fixed stack is smooth with tangent bundle ${{\mathcal T}}^{1, f}$ . Hence, $[ {\overline {\mathfrak M}}_{\Gamma }]^{\mathrm {vir}} = [ {\overline {\mathfrak M}}_{\Gamma }]$ and $[\overline {M}_{\Gamma }]^{\mathrm {vir}} = [\overline {M}_{\Gamma }]$ . By (5.14), we obtain

$$ \begin{align*} \langle \rangle_{1, d\beta_3} \quad = \quad \sum_{\Gamma} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align*} $$

In view of the splitting (5.13), the invariant $\langle \rangle _{1, d\beta _3}$ can be written as

(5.16) $$ \begin{align} \sum_{1 \leq i\neq j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} \,\, + \,\, \sum_{i=1}^{\chi(X)} \,\,\sum_{\Gamma \in {\mathcal T}_{d,i}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )} \end{align} $$

5.3 Reformulation of $ \sum _{1 \leq i \neq j \leq \chi (X)} \,\, \sum _{\Gamma \in {{ \mathcal S}}_{d,i,j}}$

In this subsection, we will reformulate the summation $\sum _{1 \leq i\neq j \leq \chi (X)} \,\, \sum _{\Gamma \in {{\mathcal S}}_{d,i,j}}$ in (5.16) by a suitable genus- $1$ Gromov-Witten invariant of $X \times {X^{[2]}}$ . It allows us to reduce the computation of $\langle \rangle _{1, d\beta _3}$ to the local affine charts $U_i \ni x_i$ .

For $1 \le i \le \chi (X)$ and $1 \le k \le 2$ , let $R_{i,i}^{(k)} = (x_i, \xi _{i, k}) \in X \times {X^{[2]}}$ and

$$ \begin{align*}C_{i, i} = \{ x_i \} \times M_2(x_i) \subset X \times {X^{[2]}}. \end{align*} $$

For $1 \le i \ne j \le \chi (X)$ , regard the curve $C_{i, j} \subset X^{[3]}$ in Lemma 5.2 (i) as the curve $\{ x_j \} \times M_2(x_i) \subset X \times {X^{[2]}}$ . The $ {\mathbb T}$ -action on X induces a $ {\mathbb T}$ -action on $X \times {X^{[2]}}$ . The $ {\mathbb T}$ -fixed point set $(X \times {X^{[2]}})^{{\mathbb T}}$ consists of the points $R_{i,j}^{(k)}$ with $1 \le i, j \le \chi (X)$ and $1 \le k \le 2$ . The $ {\mathbb T}$ -invariant curves in $X \times {X^{[2]}}$ contracted by

$$ \begin{align*}\mathrm{Id} \times \rho_2: \,\, X \times {X^{[2]}} \to X \times X^{(2)} \end{align*} $$

are precisely the curves $C_{i, j}$ with $1 \le i, j \le \chi (X)$ . The decompositions of the tangent spaces of $X \times {X^{[2]}}$ at the points $R_{i,j}^{(k)}$ are given by the right-hand sides of (5.5) and (5.6). So we keep using $T_{R_{i,j}^{(k)}}$ to denote the tangent space of $X \times {X^{[2]}}$ at $R_{i,j}^{(k)}$ . Similarly, if $f: {\mathbb P}^1 \to X \times {X^{[2]}}$ is a degree-d morphism such that $f({\mathbb P}^1) = C_{i,j}$ and f is totally ramified at the two $ {\mathbb T}$ -fixed points in $f({\mathbb P}^1)$ , then the Euler characteristic $\chi (f^*T_{X \times {X^{[2]}}})$ is given by the right-hand side of (5.10).

Regard $\beta _2 \in H_2( {X^{[2]}})$ as in $H_2(X \times {X^{[2]}})$ . Apply localization to the moduli space

$$ \begin{align*}{\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2) \end{align*} $$

whose expected dimension is equal to $0$ . The connected components of the $ {\mathbb T}$ -fixed point set $\big ( {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2) \big )^{{\mathbb T}}$ are indexed by stable graphs $\Gamma $ . For $1 \le i, j \le \chi (X)$ , let ${{\mathcal S}}_{d,i,j}$ be the set of all stable graphs $\Gamma $ such that $f(C) = C_{i,j}$ for every stable map $[f: C \to X \times {X^{[2]}}]$ in the connected component $ {\overline {\mathfrak M}}_{\Gamma }$ indexed by $\Gamma $ . Note that when $i \ne j$ , ${{\mathcal S}}_{d,i,j}$ can be identified with the set ${{\mathcal S}}_{d,i,j}$ introduced in (5.13). Moreover, for $\Gamma \in {{\mathcal S}}_{d,i,j}$ with $i \ne j$ , $ {\overline {\mathfrak M}}_{\Gamma }$ can be identified with the connected component $ {\overline {\mathfrak M}}_{\Gamma }$ introduced in (5.11). By the virtual localization formula,

(5.17) $$ \begin{align} \int_{[ {\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2)]^{\mathrm{vir}} } 1 \quad = \quad \sum_{1 \leq i, j \leq \chi(X)} \,\, \sum_{\Gamma \in {{\mathcal S}}_{d,i,j}} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Note that for each graph $\Gamma \in {{\mathcal S}}_{d,i,j}$ with $i \ne j$ , the summand $\displaystyle {\frac {1}{|\mathbf {A}_{\Gamma }|} \int _{[\overline {M}_{\Gamma }]} \frac {1}{e(N_{\Gamma }^{\mathrm {vir}} )}}$ in (5.17) is equal to the corresponding summand in (5.16).

Proof. We have $ {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2) \cong X \times {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . By the results in [Reference Hu, Li and Qin15], the moduli space $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ is smooth (as a stack) with dimension $(2d+2)$ , and the obstruction sheaf $\mathcal Ob = R^1(f_{1,0})_*\mathrm {ev}_1^*T_{ {X^{[2]}}}$ on $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ is locally free of rank $(2d+2)$ where $f_{1,0}$ (respectively, $\mathrm {ev}_1$ ) denotes the forgetful (respectively, evaluation) map on $ {\overline {\mathfrak M}}_{1, 1}( {X^{[2]}}, d\beta _2)$ . Moreover, we have

(5.18) $$ \begin{align} \langle \rangle_{1, d\beta_2} = \deg c_{2d+2}(\mathcal Ob) = \frac{1}{12d} \cdot K_X^2. \end{align} $$

Let $\phi _1$ and $\phi _2$ be the two projections on $X \times {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . Let $\mathcal H_1$ be the (rank- $1$ ) Hodge bundle over the moduli space $ {\overline {\mathfrak M}}_{1, 0}( {X^{[2]}}, d\beta _2)$ . A direct computation shows that the obstruction sheaf over $ {\overline {\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta _2)$ is isomorphic to

$$ \begin{align*}(\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}) \oplus \phi_2^*\mathcal Ob \end{align*} $$

which is locally free of rank $(2d+4)$ . Therefore, we conclude that

$$ \begin{align*} [ {\overline{\mathfrak M}}_{1, 0}(X \times {X^{[2]}}, d\beta_2)]^{\mathrm{vir}} &= c_{2d+4}\big ((\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}) \oplus \phi_2^*\mathcal Ob \big ) \\ &= c_2\big (\phi_1^*T_X \otimes \phi_2^*\mathcal H_1^{\vee}\big ) \cdot c_{2d+2}\big (\phi_2^*\mathcal Ob \big ). \end{align*} $$

Combining this with (5.18), we immediately verify our lemma.

From (5.16), (5.17) and Lemma 5.3, we conclude that

(5.19) $$ \begin{align} \langle \rangle_{1, d\beta_3} = \frac{1}{12d} \cdot \chi(X) \cdot K_X^2 \,\, + \,\, \sum_{i=1}^{\chi(X)} \,\, \left (\sum_{\Gamma \in {\mathcal T}_{d,i}} - \sum_{\Gamma \in {\mathcal S}_{d,i, i}} \right ) \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

Note that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ depends only on the local chart $U_i \ni x_i$ .

For simplicity, whenever S is a set of stable graphs, we use $\sum _{\Gamma \in S}$ to denote

(5.20) $$ \begin{align} \sum_{\Gamma \in S} \frac{1}{|\mathbf{A}_{\Gamma}|} \int_{[\overline{M}_{\Gamma}]} \frac{1}{e(N_{\Gamma}^{\mathrm{vir}} )}. \end{align} $$

5.4 A reduction lemma

In this subsection, we will prove a reduction lemma which indicates that we may ignore most of the stable graphs in ${\mathcal T}_{d,i}$ and ${\mathcal S}_{d,i, i}$ when we evaluate the summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ in (5.19).

Before we state the reduction lemma, we present the motivations. As we will see in the next two subsections, $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{p_{1,1}(w_i, z_i)}{q_{1,1}(w_i, z_i)} + (w_i + z_i)^3 \cdot \frac{p_{1,2}(w_i, z_i)}{q_{1,2}(w_i, z_i)} \end{align*} $$

where $p_{1,1}(w_i, z_i), q_{1,1}(w_i, z_i), p_{1,2}(w_i, z_i), q_{1,2}(w_i, z_i) \in {\mathbb Q}[w_i, z_i]$ are symmetric homogeneous polynomials independent of i and X, $(w_i + z_i) \nmid q_{1,1}(w_i, z_i)$ , $(w_i + z_i) \nmid q_{1,2}(w_i, z_i)$ , $\deg (q_{1,1}) = \deg (p_{1,1}) + 2$ , $\deg (q_{1,2}) = \deg (p_{1,2}) + 3$ and all the roots of $q_{1,1}(w, 1)$ and $q_{1,2}(w, 1)$ are rational. Note that $p_{1,1}(w_i, z_i), q_{1,1}(w_i, z_i), p_{1,2}(w_i, z_i)$ and $q_{1,2}(w_i, z_i)$ can be expressed as polynomials in $w_i + z_i$ and $w_iz_i$ . So the summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ can be rewritten as

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{\tilde a \cdot (w_iz_i)^m}{q_{1,1}(w_i, z_i)} + (w_i + z_i)^3 \cdot \frac{p_{2,2}(w_i, z_i)}{q_{1,2}(w_i, z_i)} \end{align*} $$

where $\tilde a$ and m are independent of i and X, and $p_{2,2}(w_i, z_i)$ is a symmetric homogeneous polynomial independent of i and X. Put $q_{1,1}(w_i, z_i) = \tilde a_0 (w_iz_i)^{m+1} + \tilde a_1 (w_iz_i)^{m}(w_i + z_i)^2 + \ldots + \tilde a_{m+1}(w_i + z_i)^{2(m+1)}$ . Then,

$$ \begin{align*}(w_i + z_i)^2 \cdot \frac{\tilde a \cdot (w_iz_i)^m}{q_{1,1}(w_i, z_i)} = a \cdot \frac{(w_i + z_i)^2}{w_i z_i} - (w_i + z_i)^4 \cdot \frac{p_{2,1}(w_i, z_i)}{w_iz_i \cdot q_{1,1}(w_i, z_i)} \end{align*} $$

where $a = \tilde a/\tilde a_0$ , and $p_{2,1}(w_i, z_i)$ is a symmetric homogeneous polynomial independent of i and X. It follows that $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

(5.21) $$ \begin{align} a_d \cdot \frac{(w_i + z_i)^2}{w_i z_i} + (w_i + z_i)^3 \cdot \frac{p_d(w_i, z_i)}{q_d(w_i, z_i)} \end{align} $$

where $a_d (= a), p_d(w_i, z_i)$ and $q_d(w_i, z_i)$ are independent of i and X and depend only on d, $a_d \in {\mathbb Q}$ , $p_d(w_i, z_i)$ and $q_d(w_i, z_i)$ are symmetric homogeneous polynomials in ${\mathbb Q}[w_i, z_i]$ , $(w_i + z_i) \nmid q_d(w_i, z_i)$ and the roots of $q_d(w, 1)$ are rational. Our reduction lemma below asserts that $p_d = 0$ .

Lemma 5.4. The summation $\sum _{\Gamma \in {\mathcal T}_{d,i}} - \sum _{\Gamma \in {\mathcal S}_{d,i, i}}$ is of the form

(5.22) $$ \begin{align} a_d \cdot \frac{(w_i + z_i)^2}{w_i z_i} \end{align} $$

where $a_d \in {\mathbb Q}$ is independent of i and X and depends only on d, and

(5.23) $$ \begin{align} \langle \rangle_{1, d\beta_3} = \left ( a_d + \frac{1}{12d} \cdot \chi(X) \right ) K_X^2. \end{align} $$

Proof. Note that (5.23) follows from (5.19), (5.22) and (5.1). In the following, we will prove (5.22) (i.e., we will show that $p_d = 0$ in (5.21)). For convenience, we will simply write $a, p, q$ instead of $a_d, p_d, q_d$ . Assume $p \ne 0$ . We will draw contradictions. We may further assume that $p(w_i, z_i)$ and $q(w_i, z_i)$ have no common factors of positive degrees and that $q(w, 1)$ is monic.

First of all, we conclude from (5.19), (5.21) and (5.1) that

(5.24) $$ \begin{align} \sum_{i=1}^{\chi(X)} (w_i + z_i)^3 \cdot \frac{p(w_i, z_i)}{q(w_i, z_i)} = \langle \rangle_{1, d\beta_3} - \frac{1}{12d} \cdot \chi(X) \cdot K_X^2 - aK_X^2. \end{align} $$

For simplicity, denote the right-hand side of (5.24) by $e(X, d)$ . The symmetric polynomials $p(w_i, z_i)$ and $q(w_i, z_i)$ can be expressed as polynomials in $(w_i + z_i)$ and $w_iz_i$ . Since $(w_i + z_i) \nmid q(w_i, z_i)$ , $q(w_i, z_i)$ is of the form

(5.25) $$ \begin{align} q(w_i, z_i) = (w_iz_i)^{n_0} \cdot \tilde q(w_i, z_i) = (w_iz_i)^{n_0} \cdot \prod_{j=1}^k \big ( (w_i + z_i)^2 + a_j w_i z_i \big )^{n_j} \end{align} $$

where $n_0 \ge 0$ , $k \ge 0$ , $a_1, \ldots , a_k$ are distinct and $a_j \ne 0$ and $n_j> 0$ for every j. So $\deg (q)$ is even, $\deg (p) = \deg (q) - 3$ is odd and $(w_i + z_i)|p(w_i, z_i)$ . Put

$$ \begin{align*}p(w_i, z_i) = (w_i + z_i) \cdot \tilde p(w_i, z_i). \end{align*} $$

Being of even degree, the symmetric homogeneous polynomial $\tilde p(w_i, z_i)$ is a polynomial of $(w_i + z_i)^2$ and $w_i z_i$ . By (5.24), we have

(5.26) $$ \begin{align} \sum_{i=1}^{\chi(X)} (w_i + z_i)^4 \cdot \frac{\tilde p(w_i, z_i)}{q(w_i, z_i)} = e(X, d). \end{align} $$

For $X = {\mathbb P}^2$ and ${\mathbb P}^1 \times {\mathbb P}^1$ , the weights $w_i$ and $z_i$ are of the form

$$ \begin{align*} & \{(w_i, z_i)| 1 \le i \le \chi(X) \} \\ =&\begin{cases} \{(w,z), (w-z, -z), (z-w, -w) \},&\mbox{if } X = {\mathbb P}^2 \\ \{(w,z), (-w, -z), (w, -z), (-w, z) \},& \mbox{if } X = {\mathbb P}^1 \times {\mathbb P}^1. \end{cases} \end{align*} $$

Set $z=1$ . Letting $X = {\mathbb P}^2$ and $X = {\mathbb P}^1 \times {\mathbb P}^1$ in (5.26), respectively, we obtain

(5.27) $$ \begin{align} (w + 1)^4 \, \frac{\tilde p(w, 1)}{q(w, 1)} + (w-2)^4 \, \frac{\tilde p(w-1, -1)}{q(w-1, -1)} + (1-2w)^4 \, \frac{\tilde p(1-w, -w)}{q(1-w, -w)} =e_1, \end{align} $$

(5.28) $$ \begin{align} (w + 1)^4 \cdot \frac{\tilde p(w, 1)}{q(w, 1)} + (-w + 1)^4 \cdot \frac{\tilde p(-w, 1)}{q(-w, 1)} =e_2 \end{align} $$

where $e_1=e({\mathbb P}^2, d)$ and $e_2 = e({\mathbb P}^1 \times {\mathbb P}^1, d)/2$ . Since $p(w_i, z_i)$ and $q(w_i, z_i)$ have no common factor of positive degree, neither do $\tilde p(w, 1)$ and $\tilde q(w, 1)$ . If $k \ge 1$ , then by (5.28) and (5.25), $\tilde q(w, 1)|\tilde q(-w, 1)$ . So $\tilde q(w, 1) = \tilde q(-w, 1)$ since they are monic and $\tilde q(w_i, z_i) = \tilde q(-w_i, z_i)$ . Since the roots of $q(w, 1)$ are rational, $a_j \ne -2$ and

$$ \begin{align*}(w_i + z_i)^2 + a_j w_i z_i \ne (w_i - z_i)^2 - a_j w_i z_i \end{align*} $$

for j and i. Therefore, $(w_i + z_i)^2 + a_j w_i z_i$ and $(w_i - z_i)^2 - a_j w_i z_i$ are distinct factors in the decomposition (5.25) of $q(w_i, z_i)$ , and $q(w_i, z_i) $ can be rewritten as

(5.29) $$ \begin{align} & (w_iz_i)^{n_0} \cdot \prod_{j=1}^s \big ( ((w_i + z_i)^2 + a_j w_i z_i) ((w_i - z_i)^2 - a_j w_i z_i) \big )^{n_j} \end{align} $$

(5.30) $$ \begin{align} =&(w_iz_i)^{n_0} \cdot \prod_{j=1}^s \big ((w_i^2 + z_i^2)^2 - \tilde a_j (w_i z_i)^2 \big )^{n_j} \end{align} $$

where $s = k/2 \ge 0$ , $\tilde a_j = (2+a_j)^2$ and $\tilde a_1, \ldots , \tilde a_s$ are distinct. Since the roots of $(w +1)^2 + a_j w$ are rational, $\tilde a_j \ge 4$ . Since $(w_i + z_i) \nmid q(w_i, z_i)$ , $\tilde a_j \ne 4$ . So $\tilde a_j> 4$ , and $a_j \ne 0, -4$ . Let $n = \deg (q) = 2n_0 + 4(n_1 + \ldots + n_s).$ Then $\deg (\tilde p) = n-4$ .

If $n_0$ is positive and even, then as a polynomial in $(w_i + z_i)^2$ and $w_i z_i$ , $\tilde p(w_i, z_i)$ contains the monomial $(w_i + z_i)^{n - 4}$ with nonzero coefficient. So $(w + 1)^4 \, \tilde p(w, 1)$ is a polynomial of degree n in w. Since $q(w, 1)$ is of degree $n - n_0$ , letting $w \to \infty $ in (5.28), we get $\infty = e_2$ . This is impossible since $e_2$ is a finite number.

If $n_0$ is odd with $n_0 \ge 3$ , then write $\tilde p(w_i, z_i) = \sum _{j=0}^{n-4} h_j w_i^j z_i^{(n-4)-j}$ . Since $\tilde p(w_i, z_i)$ is symmetric, $h_j = h_{(n-4)-j}$ . Since $(w_iz_i) \nmid \tilde p(w_i, z_i)$ , $h_0 \ne 0$ . Since $a_j \not \in \{0, -4\}$ , we see that $w \nmid q(w-1, -1)$ and $w \nmid \tilde q(1-w, -w)$ . Substitute (5.30) into (5.27) and (5.28). Expanding the left-hand sides of (5.27) and (5.28), we get

$$ \begin{align*} \big ( (n_0 + 8)h_0 + 2h_1 \big )w^{-(n_0-1)} + O(w^{-(n_0-2)}) &= e_1, \\ (8 h_0 + 2h_1)w^{-(n_0-1)} + O(w^{-(n_0-3)}) &= e_2 \end{align*} $$

where $O(w^{-i})$ with $i> 0$ denotes a term such that as $w \to 0$ , $|O(w^{-i})| \le c |w^{-i}|$ for some constant c. The two coefficients of $w^{-(n_0-1)}$ cannot be $0$ simultaneously. So letting $w \to 0$ , we have either $\infty = e_1$ or $\infty = e_2$ . This is absurd.

By the previous two paragraphs, $n_0 = 0$ or $1$ . Since $\deg (q) \ge 3$ , $s \ge 1$ . The roots of $(w^2 + 1)^2 - \tilde a_j w^2$ are $\alpha , \alpha ^{-1}, -\alpha , -\alpha ^{-1}$ for some rational number $\alpha \ne 0, 1$ , and these four roots are mutually distinct. Let $\alpha _0, \alpha _0^{-1}, -\alpha _0, -\alpha _0^{-1}$ be the roots of $(w^2 + 1)^2 - \tilde a_1 w^2$ . By symmetry, let $0 < \alpha _0 < 1$ . If $(w + \alpha _0) \nmid \big (q(w-1,-1)q(1-w, -w) \big )$ , then letting $w \to -\alpha _0$ in (5.27), we obtain the contradiction $\infty = e_1$ . If $(w + \alpha _0)|q(w-1,-1)$ , then $(\alpha _0 + 1) \ne 0$ is a root of $q(w, 1)$ . Therefore, $1/(\alpha _0 + 1)$ is a root of $q(w, 1)$ as well. Similarly, if $(w + \alpha _0)|q(1-w, -w)$ , then $-(\alpha _0 + 1)/\alpha _0$ is a root of $q(w, 1)$ ; in this case, $\alpha _0/(\alpha _0 + 1)$ is also a root of $q(w, 1)$ . Note that $0 < 1/(\alpha _0 + 1), \alpha _0/(\alpha _0 + 1) < 1$ . Define two functions

$$ \begin{align*}\phi_1(x) = 1/(x + 1), \quad \phi_2(x) = x/(x + 1). \end{align*} $$

So there exists $\psi _1 \in \{\phi _1, \phi _2\}$ such that $\psi _1(\alpha _0)$ is a root of $q(w, 1)$ . Putting $\alpha _1 = \psi _1(\alpha _0)$ and repeating the above process, we see that $q(w, 1)$ has a sequence of roots

$$ \begin{align*}\alpha_k = \psi_k \cdots \psi_1(\alpha_0), \quad k \ge 1 \end{align*} $$

where $\psi _1, \ldots , \psi _k \in \{\phi _1, \phi _2\}$ . By induction, we get $0 < \alpha _k < 1$ for every $k \ge 0$ .

Claim. $\alpha _i \ne \alpha _k$ whenever $i, k \ge 0$ and $i \ne k$ .

Proof. Assume $\alpha _i = \alpha _k$ with $0 \le i < k$ . Then, $\alpha _k = \psi _k \cdots \psi _{i+1}(\alpha _i)$ . So we may assume that $i = 0$ , $\alpha _0 = \alpha _k$ and $\alpha _k = \psi _k \cdots \psi _1(\alpha _0)$ . Since $\psi _1, \ldots , \psi _k \in \{\phi _1, \phi _2\}$ , we see from induction that $\alpha _k = (a \alpha _0 + b)/(c \alpha _0 + d)$ for some integers $a \ge 0, b \ge 0, c \ge 1, d \ge 1$ satisfying $ad - bc = \pm 1$ . So $\alpha _0 = (a \alpha _0 + b)/(c \alpha _0 + d)$ , and we get

$$ \begin{align*}c \alpha_0^2 + (d-a) \alpha_0 - b = 0. \end{align*} $$

Since $\alpha _0$ is a rational number, $(d-a)^2 + 4bc = f^2$ for some integer f. If $ad - bc = -1$ , then $f^2 = (d+a)^2 + 4$ , and so $d + a = 0$ , which contradicts $a \ge 0$ and $d \ge 1$ . If $ad - bc = 1$ , then $f^2 + 4 = (d+a)^2$ , and so $f=0$ and $d + a = 2$ . Since $a \ge 0, b \ge 0, c \ge 1, d \ge 1$ are integers satisfying $ad - bc = 1$ , we must have $a = d = 1$ , $b = 0$ and $\alpha _0 = 0$ . This contradicts $\alpha _0 \ne 0$ .

We continue the proof of our lemma. By the above claim, the polynomial $q(w, 1)$ has infinitely many roots $\alpha _k, k \ge 0$ which are mutually distinct. This is absurd.