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.
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