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.

2000 AMS Subject Classification:

• Primary 14K10
• 14K25
• Secondary 14M25
• 14T05

Keywords:

• abelian varieties
• moduli spaces
• theta functions
• toric varieties

ACKNOWLEDGMENTS

We thank Melody Chan, Igor Dolgachev, Jan Draisma, Bert van Geemen, Sam Grushevsky, Thomas Kahle, Daniel Plaumann, and Riccardo Salvati Manni for helpful communications. We made extensive use of the software packages Sage, Macaulay2, and GAP. Qingchun Ren was supported by a Berkeley fellowship. Gus Schrader was supported by a Fulbright fellowship. Steven Sam was supported by an NDSEG fellowship and a Miller research fellowship. Bernd Sturmfels was partially supported by NSF grant DMS-0968882.

Notes

¹In July 2013, Bert van Geemen posted a new article [van Geemen 13], which constructs further elements of the ideal , thus disproving our Conjecture 8.6.

²These files can be found at http://math.berkeley.edu/svs/supp/univ_kummer/ .