Enumerative tropical algebraic geometry in R2

Abstract

The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon. The formula was announced earlier in http://arxiv.org/abs/math.AG/0209253. The result is established with the help of the so-called tropical algebraic geometry. This geometry allows one to replace complex toric varieties with the Euclidean n-space and holomorphic curves with certain piecewise-linear graphs there.

83 pages, 20 figures, Version 4, to appear in the Journal of the AMS

References in corpus (2)

• Correspondence principle for idempotent calculus and some computer applications
• Welschinger invariant and enumeration of real plane rational curves

Cited by in corpus (39)

• Computing Tropical Varieties
• On the tropical Torelli map
• Torelli theorem for graphs and tropical curves
• Floor decompositions of tropical curves : the planar case
• Logarithmic equivalence of Welschinger and Gromov-Witten invariants
• Mirror symmetry for toric Fano manifolds via SYZ transformations
• The number of extreme points of tropical polyhedra
• Zero distribution of random sparse polynomials
• Geometry of tropical moduli spaces and linkage of graphs
• Tropical matrix duality and Green's D relation
• The number of realizations of a Laman graph
• Faithful realizability of tropical curves
• Enumeration of rational curves with cross-ratio constraints
• Stable maps to Looijenga pairs
• Tropical curves in sandpiles
• Tropical Lagrangian Hypersurfaces are Unobstructed
• Tropical quantum field theory, mirror polyvector fields, and multiplicities of tropical curves
• On the log-local principle for the toric boundary
• Tautological relations in Hodge field theory
• A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice
• Generic Tropical Varieties
• Integration over Tropical Plane Curves and Ultradiscretization
• The Orbifold Topological Vertex
• Refined floor diagrams from higher genera and lambda classes
• Balancing conditions in global tropical geometry
• Tropicalization and irreducibility of Generalized Vandermonde Determinants
• Rational quintics in the real plane
• Enumerative geometry of elliptic curves on toric surfaces
• Haas' theorem revisited
• Mirror symmetry for the Tate curve via tropical and log corals
• Military Simulator - A Case Study of Behaviour Tree and Unity based architecture
• String networks as tropical curves
• Second main theorem with tropical hypersurfaces and defect relation
• Combinatorics of tropical Hurwitz cycles
• Tropical spectral curves, Fay's trisecant identity, and generalized ultradiscrete Toda lattice
• The Nullstellensatz and Positivstellensatz for Sparse Tropical Polynomial Systems
• Polynomiality properties of tropical refined invariants
• The tropical discriminant of a polynomial map on a plane
• Idempotent and tropical mathematics. Complexity of algorithms and interval analysis