On the Coble quartic and Fourier–Jacobi expansion of theta relations

In [Q. Ren, S. Sam, G. Schrader and B. Sturmfels, The universal Kummer threefold, Experiment Math.22(3) (2013) 327–362], the authors conjectured equations for the universal Kummer variety in genus 3 case. Although, most of these equations are obtained from the Fourier–Jacobi expansion of relations among theta constants in genus 4, the more prominent one, Coble's quartic, cf. [A. Coble, Algebraic Geometry and Theta Functions, American Mathematical Society Colloquium Publications, Vol. 10 (American Mathematical Society, 1929)] was obtained differently, cf. [S. Grushevsky and R. Salvati Manni, On Coble's quartic, preprint (2012), arXiv:1212.1895] too. The aim of this paper is to show that Coble's quartic can be obtained as Fourier–Jacobi expansion of a relation among theta-constants in genus 4. We get also one more relation that could be in the ideal described in [Experiment Math.22(3) (2013) 327–362].