Federico Ardila ; Florian Block - Universal Polynomials for Severi Degrees of Toric Surfaces

dmtcs:3089 - Discrete Mathematics & Theoretical Computer Science, January 1, 2012, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) - https://doi.org/10.46298/dmtcs.3089
Universal Polynomials for Severi Degrees of Toric Surfaces

Authors: Federico Ardila ; Florian Block

    The Severi variety parametrizes 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 $\mathbb{CP}^2$. 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". 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.


    Volume: DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)
    Section: Proceedings
    Published on: January 1, 2012
    Imported on: January 31, 2017
    Keywords: Enumerative geometry, toric surfaces, Gromov-Witten theory, Severi degrees, node polynomials,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
    Fundings :
      Source : OpenAIRE Research Graph
    • Combinatorics in Geometry; Funder: National Science Foundation; Code: 0801075
    • CAREER: Matroids, polytopes, and their valuations in algebra and geometry; Funder: National Science Foundation; Code: 0956178

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    Source : ScholeXplorer IsReferencedBy ARXIV 1105.2004
    Source : ScholeXplorer IsReferencedBy DOI 10.1112/s0010437x13007409
    Source : ScholeXplorer IsReferencedBy DOI 10.48550/arxiv.1105.2004
    • 1105.2004
    • 10.1112/s0010437x13007409
    • 10.1112/s0010437x13007409
    • 10.1112/s0010437x13007409
    • 10.48550/arxiv.1105.2004
    Genus 0 characteristic numbers of the tropical projective plane
    Bertrand, Benoît ; Brugalle, Erwan ; Mikhalkin, Grigory ;

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