论文标题
$ g $ -INVARIANT HILBERT计划在Abelian表面和Orbifold Kummer表面的枚举几何形状上
$G$-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface
论文作者
论文摘要
对于有限组$ g $具有符号操作的Abelian Surface $ a $，可以定义$ g $ -Invariant Hilbert Schemes \ [z_ {z_ {a，g}（q）= \ sum_ = \ sum_ {d = 0}^{\ sum_ e（\ text {hilb}^{d}（a）^{g}）q^{d}。并在ETA产品方面给出明确的表达。还给出了$χ_{y} $ - $ \ text {hilb}（a）^{g} $的精制公式。对于由标准互动生成的组$τ：a \至$的组，我们的公式来自Orbifold Kummer Surface $ [A/τ] $的枚举几何形状。我们证明，堆栈中曲线的虚拟计数由$χ_{y}（\ text {hilb}（a）^τ）$控制。此外，$ z_ {a，τ} $的系数是正确的（加权）有理曲线的计数，与Bryan，Oberdieck，Pandharipande和Yin的过度计数一致。
For an Abelian surface $A$ with a symplectic action by a finite group $G$, one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A, G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove the reciprocal $Z_{A,G}^{-1}$ is a modular form of weight $\frac{1}{2}e(A/G)$ for the congruence subgroup $Γ_{0}(|G|)$, and give explicit expressions in terms of eta products. Refined formulas for the $χ_{y}$-genera of $\text{Hilb}(A)^{G}$ are also given. For the group generated by the standard involution $τ: A \to A$, our formulas arise from the enumerative geometry of the orbifold Kummer surface $[A/τ]$. We prove that a virtual count of curves in the stack is governed by $χ_{y}(\text{Hilb}(A)^τ)$. Moreover, the coefficients of $Z_{A, τ}$ are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.
