The Universal Kummer Threefold

Abstract

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