Universal Polynomials for Severi Degrees of Toric Surfaces

TL;DR

This paper proves that Severi degrees for a broad class of toric surfaces are eventually polynomial functions of multidegree and surface parameters, using tropical geometry and combinatorial methods.

Contribution

It extends the polynomiality of Severi degrees to toric surfaces and introduces new methods involving floor diagrams and polytope volume analysis.

Findings

Severi degrees are polynomial in multidegree for large values.

Severi degrees are polynomial as a function of the surface.

Explicit computations for Hirzebruch and singular surfaces demonstrate the results.

Abstract

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 coincide with the Gromov-Witten invariants of P^2. 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 large family of 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 and of a singular surface. Our strategy is to use tropical geometry to express Severi degrees in terms of Brugalle 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.