  The solutions to the Kadomtsev-Petviashvili equation that arise from a fixed
complex algebraic curve are parametrized by a threefold in a weighted
projective space, which we name after Boris Dubrovin. Current methods from
nonlinear algebra are applied to study parametrizations and defining ideals of
Dubrovin threefolds. We highlight the dichotomy between transcendental
representations and exact algebraic computations. Our main result on the
algebraic side is a toric degeneration of the Dubrovin threefold into the
product of the underlying canonical curve and a weighted projective plane.
