The Universal Kummer Threefold

TL;DR

This paper explores the universal Kummer threefold, a complex geometric object, by computing defining equations across different parameter spaces using advanced algebraic and computational techniques.

Contribution

It provides explicit defining polynomials for the universal Kummer threefold over various moduli spaces, integrating classical theories with modern computational methods.

Findings

Derived explicit equations over the Satake hypersurface

Computed defining polynomials over the G"opel variety

Applied symbolic, numerical, toric, and tropical methods

Abstract

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in 7-dimensional projective space. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the G\"opel variety, and over the reflection representation of type E7. We develop classical themes such as theta functions and Coble's quartic hypersurface using current tools from combinatorics, geometry, and commutative algebra. Symbolic and numerical computations for genus 3 moduli spaces appear alongside toric and tropical methods.