The Universal Kummer Threefold

The universal Kummer threefold is a 9-dimensional variety that represents the total space of the 6-dimensional family of Kummer threefolds in P-7. We compute defining polynomials for three versions of this family, over the Satake hypersurface, over the Gopel variety, and over the reflection representation of type E-7. 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.