Theta Surfaces

Abstract

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

On Algebraic Theta Divisors and Rational Solutions of the KP Equation

“…He also gave a complete classification of the types of theta divisors that are algebraic in [9]. Some of Eiesland's examples are reconsidered in [2]. In this paper, we will continue this study of algebraic theta divisors and theta functions in all dimensions.…”

“…Finally, the theta divisor is also an example of a double translation hypersurface: this is a hypersurface with two distinct parametrizations as Minkowski sums of curves. Their study goes back to Sophus Lie, and modern treatments can be found in [16] and in [2].…”

“…is exactly the theta divisor. This parametrization can actually be extended to any reasonably singular curve C. More precisely, if C is a Gorenstein, reduced and connected projective curve of arithmetic genus g, then one can define its generalized Jacobian Jac(C) and then construct an Abel-Jacobi map as in (2). The (closure of the) image of this map is again a divisor in the generalized Jacobian which is called the theta divisor of the singular curve C. This lifts to a hypersurface in the complex vector space H 0 (C, ω C ) * , which is cut out by an analytic equation θ(z) which we call the theta function associated to the curve C. Such theta divisors retain many properties seen in the smooth curve case.…”

Nonlinear algebra and applications

“…It would be worthwhile to discover more about possible connections of the Hirota variety with the soliton solutions [13,Example 11], also its potential link to the Dubrovin variety. In dimension 3, the Riemann theta function degenerations recover the so-called theta surfaces [10], classically the double translation surfaces [157]. For highly singular curves, these surfaces happen to be algebraic.…”

KP Solitons from Tropical Limits

“…. , c ig ) specifies a linear form c T i z = g j=1 c ij z j , just like in (2). The coefficients a = (a 1 , a 2 , .…”