The Dubrovin threefold of an algebraic curve

TL;DR

This paper studies the Dubrovin threefold associated with algebraic curves, analyzing its algebraic structure, parametrizations, and a toric degeneration into simpler geometric components.

Contribution

It introduces a detailed algebraic study of the Dubrovin threefold, including its parametrizations, defining ideals, and a novel toric degeneration into a product of a curve and a weighted projective plane.

Findings

Established a toric degeneration of the Dubrovin threefold.

Analyzed the parametrizations and defining ideals of the threefold.

Highlighted the difference between transcendental and algebraic representations.

Abstract

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.