The topology of the zero locus of a genus 2 theta function

TL;DR

This paper determines the homotopy type of the zero locus of genus 2 theta functions, showing it is homotopy equivalent to an infinite wedge of 2-spheres, extending previous work on Torelli groups.

Contribution

It computes the homotopy type of zero loci of genus 2 theta functions, revealing they are homotopy equivalent to infinite wedges of 2-spheres, a novel topological characterization.

Findings

Zero locus homotopy equivalent to infinite wedge of 2-spheres

Extends Torelli group homotopy results to theta function zero loci

Provides explicit topological description of genus 2 theta function zeros

Abstract

Mess showed that the genus 2 Torelli group $T_2$ is isomorphic to a free group of countably infinite rank by showing that genus 2 Torelli space is homotopy equivalent to an infinite wedge of circles. As an application of his computation, we compute the homotopy type of the zero locus of any classical genus 2 theta function in $\mathfrak{h}_2 \times \mathbb{C}^2$, where $\mathfrak{h}_2$ denotes rank 2 Siegel space. Specifically, we show that the zero locus of any such function is homotopy equivalent to an infinite wedge of 2-spheres.