[Paper Review] The Singular Locus of the Theta Divisor and Quadrics through a Canonical Curve

Marco Matone, Roberto Volpato|ArXiv.org|Oct 11, 2007

Algebraic Geometry and Number Theory45 references3 citations

TL;DR

This paper introduces a new section K on a canonical curve as a central tool to characterize the singular locus Θs of the theta divisor on the Jacobian. By leveraging determinantal relations of holomorphic abelian differentials and combinatorial identities, the authors derive explicit expressions for the canonical basis of holomorphic differentials and the Mumford form in terms of theta functions, and show that K generalizes the Hessian at genus 4 to arbitrary genus, providing a modular section vanishing on the Jacobian locus and special spin structures.

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.

Motivation & Objective

• To develop a new geometric approach to the singular locus Θs of the theta divisor on the Jacobian of a canonical curve.
• To identify a canonical section K on the curve that encodes the geometry of Θs and relates to special loci in the moduli space.
• To generalize the known genus 4 result (Hessian of theta function vanishing on Jacobian locus) to arbitrary genus g ≥ 4.
• To express the Mumford form and Siegel metric on the moduli space Mg explicitly in terms of the Riemann period matrix.
• To construct holomorphic sections on Mg that vanish precisely on the Jacobian locus and special spin structures, using combinatorial theta identities.

Proposed method

• Introduces a section K associated with linear dependence of quadrics through a canonical curve, defined via determinantal relations of two-fold products of holomorphic abelian differentials.
• Uses combinatorial lemmas on symmetric products and determinants to relate higher-order theta derivatives to holomorphic differentials.
• Applies Fay’s identity and Riemann’s theta relations to express the canonical basis of H⁰(KC) and the Mumford form in terms of theta functions.
• Derives the Siegel metric on the moduli space Mg from the Kodaira-Spencer map of the Bergman kernel, showing equivalence to the metric induced by the Riemann period matrix.
• Constructs a modular section k(g) of λ₁^{d_g} on Mg for even g ≥ 4, defined via products of prime forms and determinants of differentials.
• Establishes invariance of k(g) under choice of points and bases via antisymmetry and proportionality to determinants, proving independence on moduli data.

Experimental results

Research questions

• RQ1 How can the singular locus Θs of the theta divisor be characterized geometrically for arbitrary genus g ≥ 4?
• RQ2 What is the role of the section K in encoding the geometry of Θs and relating to special loci in the moduli space?
• RQ3 Can the genus 4 Hessian of the theta function be generalized to higher genus as a modular section vanishing on the Jacobian locus?
• RQ4 How can the Mumford form and the Siegel metric on Mg be explicitly expressed in terms of the Riemann period matrix?
• RQ5 What is the structure of holomorphic sections on Mg that vanish on the Jacobian locus and special spin structures?

Key findings

• The section K is a holomorphic section of a line bundle on the moduli space Mg that vanishes exactly on the Jacobian locus and on curves with even singular spin structures.
• For genus g = 4, K reduces to the Hessian of the theta function, reproducing the only previously known explicit characterization of the Jacobian locus via a modular section.
• The section k(4) = A₄ det_{i,j∈I₄} θ_{ij}(e) is a holomorphic section of λ₁³⁴ on M₄, vanishing with order 8 on the hyperelliptic locus and order 1 on the even spin locus.
• For even genus g ≥ 4, the section k(g) is independent of the choice of points and basis, and defines a holomorphic section of λ₁^{d_g} on Mg with d_g = (12−g)N_{g−3} + (g−2)[6(g−3)(g−4)+1].
• The Mumford form is explicitly expressed in terms of theta functions via the section K, and the Siegel metric on Mg is shown to be equivalent to the Kodaira-Spencer metric of the square of the Bergman kernel.
• The canonical basis of H⁰(KC) and the constant κ[ω] in the Mumford form are derived explicitly in terms of theta functions using Fay’s identity and determinantal combinatorics.