# 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}
}
• 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…
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A convex lattice polygon Delta determines a pair (S,L) of a toric surface together with an ample toric line bundle on S. The Severi degree N^{Delta,delta} is the number of delta-nodal curves in the
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We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees.The degree of the Severi variety of plane curves of degree d and δ nodes is given by a polynomial
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• 2010
We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\delta$-nodal curves in a general $\delta$-dimensional linear system is given by a universal polynomial of
According to the G¨ ottsche conjecture (now a theorem), the degree N d; of the Severi variety of plane curves of degreed with nodes is given by a polynomial ind, providedd is large enough. These
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Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative)
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In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed
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For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very
general points in the plane? The answer is easy for 6 = 1 (namely 3 (d l ) 2) but otherwise was, to my knowledge, unknown until now. We are going to develop a recursive procedure for solving such