KP equation in Symbolic, Numerical and Combinatorial Algebraic Geometry

The Kadomtsev-Petviashvili (KP) equation is a partial differential equation that describes nonlinear wave moves. It is known that algebro-geometric approaches to the KP equation provide solutions coming from a complex algebraic curve, in terms of the Riemann theta function associated with the curve. Reviewing this relation, I will introduce an algebraic object and discuss its geometric features: the so-called Dubrovin threefold of a complex algebraic curve, which parametrizes the solutions. Mentioning the relation of this threefold with the classical algebraic geometry problem, namely the Schottky problem, I will report a procedure that is via the threefold and based on numerical algebraic geometric tools, which can be used to deal with the Schottky problem from the lens of computations. I will finally focus on the geometric behaviour of the threefold when the underlying curve degenerates. This is joint work with Daniele Agostini and Bernd Sturmfels.