Moduli of curves and abelian varieties

研究者已经完成了与g属曲线模空间的几个基本不变量相关的工作。特别是他研究了模空间M(g)从单方变（对于小g）到各种一般类型的性质。最近，研究者在M(g)上的有效因子锥上发现了一系列的反例，以证明Harris-Morrison斜率猜想。在其他工作中，研究者利用曲线的模证明了正则曲线的最小分辨率猜想，并研究了自旋曲线模空间的几何分层。本课题提出了一种在曲线模空间上定义固有坐标的新技术，该技术将M(g)上的线性级数或向量束的许多问题简化为具有环面几何风格的组合问题。特别地，这种方法被期望在M(g)的有效因子的斜率上提供一个统一的界（独立于g），从而证明斜率猜想的一个弱版本。这将表明，在模空间A(g)上，任何具有足够小斜率的g维阿贝尔变体的模形式，在M(g)上消失，这将给出在所有阿贝尔变体中区分雅可比矩阵的Schottky问题的新解。在不同的方向上，研究者建议引入一种新的M(g)分层，该分层是根据某些特殊线性曲线系统的协同性来定义的。这种几何分层可以被认为是由性给出的经典M(g)分层的更微妙的模拟，其中超椭圆曲线的模拟是K3曲面的截面。一个应用是构造截面属g的极化K3曲面的模空间F(g)的双民族模型，该模型可用于描述F(g)的相交理论。另一个项目（与S. Grushevsky合作）涉及到曲线雅可比矩阵上2-函数的线性系统的研究。利用代数几何和θ函数理论的混合，研究者希望理解这种线性系统的分层，这种分层是由沿曲线的高差分变体的多重性给出的，并将它们与标准曲线的割线变体的射影几何联系起来。代数几何中的指导问题是对代数变体进行分类直至同构。对于维数为1的变化，通过考虑g属曲线的模空间M(g)来解决这个问题。这是g属曲线的通用参数空间，M(g)是维数为3g-3的代数变化。这个空间对代数几何学者和弦理论家有着极大的兴趣，过去十年在理解M(g)几何方面取得了重大进展，涉及几何、数论和物理学的思想。

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.