The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve

TL;DR

This paper investigates the geometry of the theta divisor's singular locus for canonical curves, introducing new tools and explicit formulas involving theta functions, and explores the metric structure on the moduli space of curves.

Contribution

It introduces a novel approach using determinantal relations to analyze the singular locus of the theta divisor and derives explicit theta function expressions for key geometric quantities.

Findings

Characterizes the divisor K via quadrics containing the curve

Provides explicit theta function formulas for differentials and Mumford form

Expresses the Siegel metric and volume form in terms of the period matrix

Abstract

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Theta_s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian torus defined in terms of Theta_s. It also identifies sections of line bundles on the moduli space of algebraic curves, closely related to the Mumford isomorphism, whose zero loci characterize special varieties in the framework of the Andreotti-Mayer approach to the Schottky problem, a result which also reproduces the only previously known case g=4. This new approach, based on the combinatorics of determinantal relations for two-fold products of holomorphic abelian differentials, sheds light on basic structures, and leads to the explicit expressions, in terms of theta functions, of the canonical basis of the abelian holomorphic differentials and of the constant defining the Mumford form. Furthermore, the metric on the moduli space of canonical curves, induced by the Siegel metric, which is shown to be equivalent to the Kodaira-Spencer map of the square of the Bergman reproducing kernel, is explicitly expressed in terms of the Riemann period matrix only, a result previously known for the trivial cases g=2 and g=3. Finally, the induced Siegel volume form is expressed in terms of the Mumford form.