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Refined curve counting with tropical geometry
Part of:
Projective and enumerative geometry
Published online by Cambridge University Press: 18 August 2015
Abstract
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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.
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