Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Equations of dispersionless Hirota type F(uxixj ) = 0 have been thoroughly investigated in mathematical physics and differential geometry. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional, and that the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the generic equation corresponding to the open orbit remained elusive. Here we prove that the generic 3D Hirota equation is given by the remarkable formula ϑm(τ ) = 0, τ = i Hess(u) where ϑm is any genus 3 theta constant with even characteristics and Hess(u) is the 3 × 3 Hessian matrix of a (real-valued) function u(x1, x2, x3). Thus, generic Hirota equation coincides with the equation of the genus 3 hyperelliptic divisor (to be precise, its intersection with the imaginary part of the Siegel upper half space H3). The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo Sp(6, C)-equivalence.