On the Coble quartic and Fourier-Jacobi expansion of theta relations

TL;DR

This paper demonstrates that Coble's quartic, a key equation in genus 3 Kummer varieties, can be derived from Fourier-Jacobi expansion of genus 4 theta relations, linking different approaches in algebraic geometry.

Contribution

It shows that Coble's quartic can be obtained via Fourier-Jacobi expansion from genus 4 theta relations, connecting two methods of deriving equations for Kummer varieties.

Findings

Coble's quartic derived from genus 4 theta relations

Identified an additional relation potentially in the ideal for the universal Kummer threefold

Bridged different approaches to equations of Kummer varieties

Abstract

In the paper "The universal Kummer threefold", Q. Ren, S. Sam, G. Schrader, and B. Sturmfels (arXiv:1208.1229), conjectured equations for the universal Kummer variety in genus 3 case. Though, most of these equations are obtained from the Fourier-Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic was obtained differently. The aim of the current paper is to show that Coble's quartic can be obtained as Fourier-Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in "The universal Kummer threefold".