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Counting Plane Tropical Curves via Lattice Paths in Polygons

A projective plane tropical curve is a proper immersion of a graph into the real Cartesian plane subject to some conditions such as that the images of all the edges must be lines with rational slopes. Two important combinatorial invariants of a projective plane tropical curve are its degree, d, and genus g. First, we explore Gathmann and Markwig's approach to the study of the moduli spaces of such curves and explain their proof that the number of projective plane tropical curves, counting multiplicity, passing through n = 3d + g -1 points does not depend on the choice of points, provided they are in tropical general position. This number of curves is called a Gromov-Written invariant. Second, we discuss the proof of a theorem of Mikhalkin that allows one to compute the Gromov-Written invariant by a purely combinatorial process of counting certain lattice paths.

Identiferoai:union.ndltd.org:unt.edu/info:ark/67531/metadc1873866
Date12 1900
CreatorsZhang, Yingyu
ContributorsCherry, William, Brozovic, Douglas
PublisherUniversity of North Texas
Source SetsUniversity of North Texas
LanguageEnglish
Detected LanguageEnglish
TypeThesis or Dissertation
Formativ, 51 pages, Text
RightsPublic, Zhang, Yingyu, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved.

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