Enumerative tropical algebraic geometry in R^2

By G. Mikhalkin. 2003

613 citations recorded by the source corpus.

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 this http URL 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.

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

View Paper: https://www.semanticscholar.org/paper/2d7ba8baca116b72c7db56611d8b24c23892847b

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In Algebra and Number Theory

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Citations in the corpus

Labeled floor diagrams for plane curves (2009, background): “The goal of the second phase, which can be called discretization, is to replace piecewise-linear objects of tropical geometry by purely combinatorial ones.”
On Block-Goettsche multiplicities for planar tropical curves (2012, background): “23 of [15] these relations are independent.”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “The 0- and 1-dimensional elements of the stratification of AF are called, respectively, its vertices and edges.”
Exploded Manifolds (2009, background): “For instance, in the case that X is a toric manifold, and we use the toric boundary divisors, work of Mikhalkin, in [11] and of Siebert and Nishinou in [12] translates to saying that genus zero relative Gromov-Witten invariants can be computed by considering tropical curves in ExplX.”
Refined curve counting on complex surfaces (2012, background): “Mikhalkin [55] has shown that the Severi degrees and the real enumerative invariants can be computed via tropical geometry: he introduces tropical Severi degrees and tropical Welschinger invariants by assigning Gromov–Witten and Welschinger multiplicities to tropical curves, and shows that they 4Vak”
On basic concepts of tropical geometry (2011, methodology): “Mikhalkin also considered complex tropical hypersurfaces (see [18, Sect.”
Tropical Discriminants (2005, neutral): “Volume 00, Number 0, Pages 000–000 S 0894-0347(XX)0000-0”