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The Dubrovin threefold of an algebraic curve. (English) Zbl 1465.14036

The Dubrovin threefold is an object that appears tacitly in the prominent paper [B. A. Dubrovin, Russ. Math. Surv. 36, No. 2, 11–92 (1981; Zbl 0549.58038)] on the connection between integrable systems and Riemann surfaces. Another source that mentions this threefold is a manuscript on the Schottky problem by J. B. Little [“Another relation between approaches to the Schottky problem”, Preprint, arXiv:alg-geom/9202010, Section 2]. The aim of this paper is to develop this subject from the current perspective of nonlinear algebra [M. Michałek and B. Sturmfels, Invitation to nonlinear algebra. Providence, RI: American Mathematical Society (2021; Zbl 1477.14001)]. The solutions to the Kadomtsev-Petviashvili (KP) equation that arise from a fixed complex algebraic curve are parametrized by a threefold in a weighted projective space, which the authors name after Boris Dubrovin. Current methods from nonlinear algebra are applied to study parametrizations and defining ideals of Dubrovin threefolds. The authors highlight the dichotomy between transcendental representations and exact algebraic computations. Their 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. The paper is organized as follows. Section 1 is an introduction to the subject. Section 2 deals with parametrization by abelian functions. Section 3 deals with algebraic implicitization for plane curves. This section is concerned with algebraic representations of Dubrovin threefolds. The parametrization described in section 2 will be made explicit for plane curves, in a form that is suitable for symbolic computations. Section 4 deals with transcendental implicitization. Section 5 deals with genus four and beyond. The authors here lay the foundation for future studies of these universal equations. Section 6 is devoted to degenerations.

MSC:

14H81 Relationships between algebraic curves and physics
14H42 Theta functions and curves; Schottky problem
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

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