The Severi degree is the number of plane curves of degree d and n nodes going through a fixed generic set of (the right number of) points. For large enough d, the Severi degrees equal the Gromov-Witten invariants. In 2009, Fomin and Mikhalkin proved the 1995 conjecture that for fixed n, Severi degrees are eventually polynomial in d.
We study the Severi degrees corresponding to arbitrary "h-transverse" toric surfaces. We prove the analogous result that the Severi degrees are eventually polynomial as a function of the multidegree. Perhaps more surprisingly, we prove that the Severi degrees are also eventually polynomial "as a function of the surface".
Our strategy is to use tropical geometry to express Severi degrees in terms of Mikhalkin's floor diagrams, and then study those combinatorial objects in detail. The analysis is motivated by the work of Fomin and Mikhalkin, and requires new ideas. Our method gives rise to effective computations of Severi degrees.
The talk will be self-contained, and will focus on the combinatorial aspects of this work, which is joint with Florian Block.