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 7-dimensional projective space. 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 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.

50 pages; v2: added Remark 4.3, strengthened Lemma 8.3; v3: added references and added supplementary files to source

References in corpus (1)

• Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions

Cited by in corpus (6)

• Alternating trilinear forms on a 9-dimensional space and degenerations of (3,3)-polarized Abelian surfaces
• Translating the discrete logarithm problem on Jacobians of genus 3 hyperelliptic curves with $(\ell ,\ell ,\ell )$-isogenies
• The Coble Quadric
• Multiplicative sub-Hodge structures of conjugate varieties
• Orthogonal roots, Macdonald representations, and quasiparabolic sets
• Moduli spaces of abstract and embedded Kummer varieties