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

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.

arXiv comment23 pages, 1 figure; minor revisions; to appear in Trans. Amer. Math. Soc