The Caporaso–Harris formula and plane relative Gromov-Witten invariants in tropical geometry

By Andreas Gathmann, Hannah Markwig. 2005

89 citations recorded by the source corpus.

Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d, g) of complex plane curves of degree d and genus g through 3d + g − 1 general points with the help of relative Gromov-Witten invariants. Recently, Mikhalkin has found a way to reinterpret the numbers N(d, g) in terms of tropical geometry and to compute them by counting certain lattice paths in integral polytopes. We relate these two results by defining an analogue of the relative Gromov-Witten invariants and rederiving the Caporaso–Harris formula in terms of both tropical geometry and lattice paths.

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

View Paper: https://www.semanticscholar.org/paper/81314cdcf29f7c86bd4d25435554cf4d9cf64544

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Citations in the corpus

On the tropical Torelli map (2009, background): “For instance, this program has been carried out successfully for many problems of real and complex enumerative geometry, see for example [32,40,50,24,25,11,33,7,21].”
Labeled floor diagrams for plane curves (2009, unknown): “Markwig [16].”
A Tropical Toolkit (2006, background): “The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of degenerations of varieties using projective not-necessarily-normal toric varieties.”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “Namely, Gathmann–Markwig’s count of tropical curves in the tropical version of Caporaso– Harris formula is in a strict correspondence (in the sense of modified Mikhalkin’s correspondence theorem [7], see [2], [3]) with the count of complex algebraic curves in the original Caporaso–Harris formula.”
Refined curve counting with tropical geometry (2014, background): “This follows directly from the verbatim argument in [11]; note that their argument rests on [20, Lemma 4.”
Refined curve counting on complex surfaces (2012, methodology): “This is the analogue of the tropical proof of the Caporaso–Harris recursion formula in [23], and like the original proof of Caporaso and Harris it can be viewed as a proof by degeneration.”