Refined curve counting with tropical geometry

By Florian Block, L. Gottsche. Compositio Mathematica, 2014

89 citations recorded by the source corpus.

The Severi degree is the degree of the Severi variety parametrizing plane curves of degree $d$ with ${\it\delta}$ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable $y$, which are conjecturally equal, for large $d$. At $y=1$, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a tropical description of the refined Severi degrees, in terms of a refined tropical curve count for all toric surfaces. We also refine the equivalent count of floor diagrams for Hirzebruch and rational ruled surfaces. Our description implies that, for fixed ${\it\delta}$, the refined Severi degrees are polynomials in $d$ and $y$, for large $d$. As a consequence, we show that, for ${\it\delta}\leqslant 10$ and all $d\geqslant {\it\delta}/2+1$, both refinements of Göttsche and Shende agree and equal our refined counts of tropical curves and floor diagrams.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f16efd278600d85861c26100a29dd3cbe03e9055

Citations in the corpus

• Brief introduction to tropical geometry (2015, unknown): “Göttsche [BG14] proposed a new type of multiplicities for tropical curves (a motivation for these multiplicities is provided by a Caporaso-Harris type calculation of the refined Severi degrees; the latter degrees were introduced by Göttsche in connection with [KST11]).”