Universal Polynomials for Severi Degrees of Toric Surfaces

Details Universal Polynomials for Severi Degrees of Toric Surfaces Universal Polynomials for Severi Degrees of Toric Surfaces

2010-12-23

arxiv.org/abs/1012.5305v1

Mathematics

Algebraic Geometry

28 pages, 10 figures

Scientific paper

The Severi variety parameterizes plane curves of degree d with delta nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees equal Gromov-Witten invariants. Fomin and Mikhalkin (2009) proved the 1995 conjecture that, for fixed delta, Severi degrees are eventually polynomial in d. In this paper, we study the Severi varieties corresponding to a family of possibly non-smooth toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. More surprisingly, we show that the Severi degrees are also eventually polynomial "as a function of the surface". We illustrate our theorems by explicit computing, for a small number of nodes, the Severi degree of any large enough Hirzebruch surface. Our strategy is to use tropical geometry to express Severi degrees in terms of Brugall'e and Mikhalkin's floor diagrams, and study those combinatorial objects in detail. An important ingredient in the proof is the polynomiality of the discrete volume of a variable facet-unimodular polytope.

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