Summary: UNIVERSAL POLYNOMIALS FOR
SEVERI DEGREES OF TORIC SURFACES
FEDERICO ARDILA AND FLORIAN BLOCK
Abstract. The Severi variety parameterizes plane curves of degree d with nodes.
Its degree is called the Severi degree. For large enough d, the Severi degrees coincide
with the Gromov-Witten invariants of CP2
. Fomin and Mikhalkin (2009) proved
the 1995 conjecture that for fixed , 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.
Our strategy is to use tropical geometry to express Severi degrees in terms of
BrugallŽe 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.
1. Introduction and Main Theorems
1.1. Severi degrees and node polynomials for CP2