亏格4中的Schottky理想与超Schottky理想
Schottky versus super Schottky in genus 4
- University of Pennsylvania（宾夕法尼亚大学）
- Università degli Studi di Bari Aldo Moro（巴里阿尔多·莫罗大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明了在亏格4中，包含于超Schottky理想中的Schottky理想的最小幂次为d=4，填补了该问题唯一未解决的情形，方法涉及超周期映射余微分的秩计算与退化分析。
AI中文摘要：
对超黎曼曲面及其模空间的研究促使Witten和Felder、Kazhdan和Polishchuk提出如下问题：使得Schottky理想的d次幂包含于超Schottky理想中的最小d是多少。Felder、Kazhdan和Polishchuk证明了在奇数亏格g≥5时d=g，在偶数亏格g≥4时d∈{g-1,g}；随后Y. Shen证明了在偶数亏格g≥6时d=g。因此唯一未解决的情形是g=4。本文填补了这一空白，证明在亏格4中，包含于超Schottky理想中的Schottky理想的最小幂次也是d=g=4。该问题等价于关于超周期映射的余微分的一个问题。二次奇性贡献将一个余法向量映射为一个双向量，由斜对称的(2g-2)×(2g-2)矩阵表示。问题在于找到这样的向量，使得在一般点处该矩阵具有最大秩2g-2，即在本情形中为6。我们的计算使用C×C上关于层O(a,b,c)的标准正合序列，以及退化到消失theta零点和关于Szegő核的一些事实。在具有消失theta零点的曲线上，余微分的正则化版本可由一个显式乘法映射描述，从而容易计算其秩。该秩为4。对循环三叶消失theta零点的显式变形进行一阶计算，可区分正则化余法向量的全局高斯映射恒等式。包含高斯映射的变化给出非零的第一法向符号。所得一阶形式在极限斜对称矩阵的零空间上非零，因此在邻近曲线上秩跳跃至6，表明d=4。
英文摘要：
The study of super Riemann surfaces and their moduli led Witten and Felder, Kazhdan and Polishchuk to ask what is the smallest $d$ such that the $d$-th power of the Schottky ideal is contained in the super Schottky ideal. Felder, Kazhdan and Polishchuk proved that $d=g$ in odd genus $g\geq5$ and that $d\in\{g-1,g\}$ in even genus $g\geq4$; Y.~Shen subsequently proved that $d=g$ in even genus $g\geq6$. Thus the only remaining open case was $g=4$. Here we close this gap by showing that in genus $4$ too, the smallest power of the Schottky ideal contained in the super Schottky ideal is $d=g=4$. The question is equivalent to one concerning the codifferential of the super period map. The quadratic odd contribution sends a conormal vector to a bivector, represented by a skew-symmetric $(2g-2)\times(2g-2)$ matrix. The question is to find such a vector for which, at a generic point, this matrix has maximal rank $2g-2$, or $6$ in our case. Our computation uses the standard exact sequences on $C\times C$ attached to the sheaves $\OO(a,b,c)$, together with a degeneration to a vanishing theta-null and some facts about the Szegő kernel. At a curve with a vanishing theta-null, a regularized version of the codifferential can be described by an explicit multiplication map, allowing an easy computation of the rank. This rank turns out to be $4$. A first-order calculation on an explicit deformation of a cyclic trigonal vanishing theta-null differentiates the global Gaussian-map identity for the regularized conormal bivector. Including the variation of the Gaussian map gives a nonzero first normal symbol. The resulting first-order form is nonzero on the null space of the limiting skew-symmetric matrix, so the rank jumps to $6$ on nearby curves, showing that $d=4$.
