Universal Polynomials for Severi Degrees of Toric Surfaces

The Severi variety parameterizes plane curves of degree d with δ nodes. Its degree is called the Severi degree. For large enough d, the Severi degrees coincide with the Gromov-Witten invariants of CP. Fomin and Mikhalkin (2009) proved the 1995 conjecture that for fixed δ, 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 explicitly 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 Brugallé 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.