Moduli of curves and abelian varieties

曲线模和阿贝尔簇

• 批准号: 0500747
• 负责人: Sean Keel
• 金额: --
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500747&HistoricalAwards=false
• 关键词: Moduli; curves; abelian; varieties

The investigator has done work related to several fundamental invariants of themoduli space of curves of genus g. In particular he has studied the nature of the moduli space M(g) as it changes from being a unirational variety (for small g) to a variety of general type. Recently the investigator has found a series a counterexamples to the Harris-Morrison Slope Conjecture on the cone of effective divisors on M(g). In other works, the investigator has used moduli of curves to prove the Minimal Resolution Conjecture for canonical curves and has studied geometric stratification of moduli spaces of spin curves. This project proposes a new technique of defining intrinsic coordinates on the moduli space of curves that would reduce many problems about linear series or vector bundles over M(g) to combinatorial questions having a toric geometry flavour. In particular, this approach is expected to provide a uniform bound (independent of g) on slopes of effective divisors on M(g), and thus prove a weak version of the Slope Conjecture. This would show that any modular form on the moduli space A(g) of g-dimensional abelian varieties which has sufficiently small slope, vanishes on M(g) which would give a novel solution to the Schottky problem of distinguishing Jacobians among all abelian varieties. In a different direction, the investigator proposes to introduce a new stratification of M(g) defined in terms of syzygies of certain special linear systems of curves. This geometric stratification can be thought of as amore subtle analogue of the classical stratification of M(g) given by gonalitywhere the analogue of hyperelliptic curves are sections of K3 surfaces. One application would be a construction of a birational model of the moduli space F(g) of polarized K3 surfaces of sectional genus g which could be used to describe the intersection theory of F(g). A different project (joint with S. Grushevsky) involves the study of the linear system of 2-theta functions on the Jacobian of a curve. Using a mixture of algebraic geometry and theta function theory, the investigator hopes to understand the stratification of this linear system given by multiplicities along the higher difference varieties of the curve and relate them to the projective geoemtery of secant varieties of canonical curves.The guiding problem in algebraic geometry is to classify algebraic varieties up to isomorphism. For varieties of dimension 1 this problem is approached by considering the moduli space M(g) of curves of genus g. This is the universal parameter space for curves of genus g and M(g) is an algebraic variety of dimension 3g-3. This space is of enormous interest to algebraic geometers and string theorists and the last decade has seen major progress in understanding the geometry of M(g) involving ideas from geometry, number theory and physics.

作者研究了亏格曲线的模空间的几个基本不变量，特别研究了模空间M(G)的性质，即模空间M(G)是一种单调变种(对于小g而言)，它是一种一般类型。最近，研究者在M(G)上的有效因子锥上找到了Harris-Morison Slope猜想的一系列反例。在其他工作中，研究者用曲线的模证明了正则曲线的最小分辨猜想，并研究了自旋曲线的模空间的几何分层。这个项目提出了一种在曲线的模空间上定义内蕴坐标的新技术，它将把M(G)上的线性级数或向量丛的许多问题归结为具有环状几何性质的组合问题。特别地，这种方法有望在M(G)上的有效因子的斜率上提供一个一致的界(与g无关)，从而证明Slope猜想的一个弱版本。这将表明，具有足够小斜率的g维阿贝尔簇的模空间A(G)上的任何模形式都在M(G)上消失，这将给出区分所有阿贝尔簇中的Jacobian的肖特基问题的一个新的解决方案。在不同的方向上，研究者建议引入M(G)的一种新的分层，该分层是根据某些特殊的线性曲线系统的合集定义的。这种几何分层可以看作是Gonality给出的M(G)的经典分层的更微妙的模拟，其中超椭圆曲线的模拟是K3曲面的部分。一个应用是构造分段亏格g的极化K3曲面的模空间F(G)的双调模型，该模型可用于描述F(G)的交集理论。另一个项目(与S.Grushevsky联合)涉及研究曲线雅可比上的2-theta函数的线性系统。利用代数几何和theta函数理论相结合的方法，研究人员希望了解这种线性系统的层次性，它是由曲线的高阶差值簇上的重数所给出的，并将它们与标准曲线的割簇的射影测距联系起来。代数几何中的指导问题是对代数簇进行分类，直到同构。对于1维变差，考虑亏格g的曲线的模空间M(G)，这是g亏格曲线的泛参数空间，M(G)是维3G-3的代数族。这个空间引起了代数几何学家和弦理论家的极大兴趣，在过去的十年里，人们在理解M(G)的几何方面取得了重大进展，涉及到几何、数论和物理的思想。