AI 中文总结
该研究利用阿贝尔曲面相关的Fourier–Mukai同构与二次型，证明一般(1,d)极化阿贝尔曲面中存在任意高重数奇点的曲线，还确定了d=4时Kummer型簇的孤立不动点并得到对称线丛的结构结果。
AI 中文摘要
Yoshioka已证明：给定皮卡秩为1的复阿贝尔曲面$A$，其本原极化是非主极化且型为$(1,d)$，则存在同构$Ψ:\text{Hilb}^d_A\times\tilde{A}\to M_{\tilde{A}}(0,\tilde{l},-1)$，其中$\text{Hilb}^d_A$是$A$上长度为$d$的子概形的Hilbert概形，$M_{\tilde{A}}(0,\tilde{l},-1)$是对偶阿贝尔曲面上Gieseker稳定层的模空间。具体而言，$M_{\tilde{A}}(0,\tilde{l},-1)$参数化欧拉示性数为$-1$的秩1无挠层，这些层支撑在一条曲线上，该曲线的Néron–Severi类与$A$上极化的Néron–Severi类对偶。Fourier–Mukai变换是$Ψ$的关键组成部分。\n本文中，我们利用同构$Ψ$推导阿贝尔曲面上曲线的相关信息。当$Z\text{Hilb}^d_A$中的元素是对称的（即被$A$上的逆映射$ι$固定）时，我们使用二次型计算层$Ψ(Z)$及其支撑曲线的信息。Knutsen与Lelli-Chiesa近期证明，包含在一般$(d_1,d_2)$极化阿贝尔曲面中的几何亏格为2的曲线，其任意奇点的重数至多为6，此外还有其他约束。与之相反，我们证明：对于足够大的$d$，一般$(1,d)$极化阿贝尔曲面中存在具有任意高重数奇点的曲线。此外，当$d=4$时，我们确定了$ι$作用在Kummer型簇$K_{\tilde{A}}(0,\tilde{l},-1)$上的孤立不动点。在此过程中，我们证明了对称线丛的一些结构结果：若$d$为偶数，则奇线丛的对偶仍是奇线丛。
英文摘要
It was proven by Yoshioka that given a complex abelian surface $A$ of Picard rank $1$ whose primitive polarization is non-principal of type $(1,d)$, there is an isomorphism $Ψ:\mathrm{Hilb}^d_A\times\hat{A}\to M_{\hat{A}}(0,\hat{l},-1)$ where $\mathrm{Hilb}^d_A$ is the Hilbert scheme of lenth-$d$ subschemes of $A$ and $M_{\hat{A}}(0,\hat{l},-1)$ is a moduli space of Gieseker-stable sheaves on the dual abelian surface. Specifically, $M_{\hat{A}}(0,\hat{l},-1)$ parametrizes rank $1$ torsion-free sheaves with Euler characteristic $-1$ that are supported on a curve whose Néron--Severi class is dual to that of the polarization on $A$. The Fourier--Mukai transform is a crucial component of $Ψ$. In this paper, we use the isomorphism $Ψ$ to deduce information about curves on abelian surfaces. In the case that $Z\in \mathrm{Hilb}^d_A$ is symmetric, i.e., fixed by the inverse map $ι$ on $A$, we use quadratic forms to compute information about the sheaf $Ψ(Z)$ and its supporting curve. It was recently shown by Knutsen and Lelli-Chiesa that any singularity on a curve of geometric genus $2$ contained in a general $(d_1,d_2)$-polarized abelian surface must have multiplicity at most $6$, among other constraints. In contrast, we demonstrate there are curves with singularities of arbitrarily high multiplicity contained in general $(1,d)$-polarized abelian surfaces for sufficiently large $d$. Furthermore, we identify the isolated fixed points of $ι$ in acting on the variety of Kummer type $K_{\hat{A}}(0,\hat{l},-1)$ when $d=4$. Along the way, we prove some structural results on symmetric line bundles, showing that if $d$ is even, the dual of an odd line bundle is odd.
Comments30 pages
