On the Coble quartic


We obtain a short explicit expression for the universal Coble quartic whose partial derivatives give the defining equations for the universal family of Kummer threefolds. The Coble quartic was recently determined completely in Ren, Sam, Schrader, and Sturmfels, where (Theorem 7.1a) it was computed explicitly, as a polynomial with 372060 monomials of bidegree $(28,4)$ in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to G\"opel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland, and highlights the geometry and combinatorics of syzygetic octets of characteristics, the GIT quotient for 7 points in ${\bf P}^2$, and the corresponding representations of ${\rm Sp}(6,\Bbb{F}_2)$. One new ingredient is the relationship of G\"opel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on ${\bf P}^1$.