• DOI:10.1112/S0010437X1500754X
• Corpus ID: 119601856

Refined curve counting with tropical geometry

@article{Block2014RefinedCC,
  title={Refined curve counting with tropical geometry},
  author={Florian Block and Lothar G{\"o}ttsche},
  journal={Compositio Mathematica},
  year={2014},
  volume={152},
  pages={115 - 151}
}

• Florian Block, L. Göttsche
• Published 10 July 2014
• Mathematics
• Compositio Mathematica

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.