Curves, degenerations, and Hirota varieties

The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.