Moduli of curves and abelian varieties

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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.