Universal polynomials for Severi degrees of toric surfaces

作者：Ardila Federico^*; Block Florian

来源：Advances in Mathematics, 2013, 237: 165-193.

DOI：10.1016/j.aim.2013.01.002

• Enumerative geometry
• Toric surfaces
• Gromov-Witten theory
• Seven degrees
• Node polynomials

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 CP2. Fomin and Mikhalkin (2010) [10] proved the 1995 conjecture that for fixed delta, Severi degrees are eventually polynomial in d. %26lt;br%26gt;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 %26quot;as a function of the surface%26quot;. 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. %26lt;br%26gt;Our strategy is to use tropical geometry to express Severi degrees in terms of Brugalle and Mikhalkin%26apos;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.

• 出版日期2013-4-1