On the variety associated to the ring of theta constants in genus 3

TL;DR

This paper proves that for genus 3, the map defined by theta constants is biholomorphic onto its image, extending known results from lower genera and showing the image is a normal subvariety.

Contribution

It establishes that the theta map for genus 3 is biholomorphic onto its image and that this image is a normal subvariety, extending previous results for lower genera.

Findings

The theta map for genus 3 is biholomorphic onto its image.

The image of the theta map in genus 3 is a normal subvariety.

Extension of known results from genus 2 to genus 3.

Abstract

Due to fundamental results of Igusa and Mumford the $N=2^{g-1}(2^g+1)$ even theta constants define for each genus $g$ an injective holomorphic map of the Satake compactification $X_g(4,8)=H_g/\Gamma_g[4,8]$ into the projective space $P^{N-1}$. Moreover, this map is biholomorphic onto the image outside the Satake boundary. It is not biholomorphic on the whole in the cases $g\ge 6$. Igusa also proved that in the cases $g\le 2$ this map is biholomorphic onto the image. In this paper we extend this result to the case $g=3$. So we show that the theta map $$X_3(4,8)\to P^{35}$$ is biholomorphic onto the image. This is equivalent to the statement that the image is a normal subvariety of $P^{35}$ .