Counting curves via lattice paths in polygons

By G. Mikhalkin. 2002

72 citations recorded by the source corpus.

Abstract This Note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is P 2 or P 1 × P 1 then the invariants under consideration coincide with the Gromov–Witten invariants. The formula gives a new count even in these cases, where other computational techniques are available. To cite this article: G. Mikhalkin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d0bafa319869840f096074118a74f88761b4f5d2

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry (2002, unknown): “From a referee report.”
A tropical approach to enumerative geometry (2006, methodology): “Namely, the multiplicity of a lattice path defined in [18] is equal to the sum of the multiplicities of the nodal amoebas that correspond to the subdivisions of ∆ arising from the given path in the course of the construction in [18].”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “Theorem 2 (see [6], Theorem 6, and [8], Proposition 6.”
Amoebas of Algebraic Varieties and Tropical Geometry (2004, background): “There is a combinatorial rule (see [30] or [31]) that associates a non-negative integer multiplicity to every λ-increasing lattice path of length x+g−1, i.”
The Tropical Totally Positive Grassmannian (2003, background): “We then investigate the case of the Grassmannian, denoting the resulting fan Trop+ Grk,n.”
A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces (2004, methodology): “Following the ideas of [8, 9], we intend to choose a specific configuration of the points x1, .”
Enumerative tropical algebraic geometry (2003, background): “We have Log ◦Ht = Logt .”
Lower bounds for real solutions to sparse polynomial systems (2004, background): “Welschinger discovered an invariant which gives a lower bound, and work of Mikhalkin and of Itenberg, Kharlamov, and Shustin shows that this lower bound is non-zero and in fact quite large.”
Enumerative tropical algebraic geometry in R^2 (2003, unknown): “2000 Mathematics Subject Classification.”
Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry (2003, result): “Remark 3.9. Using the results of this paragraph together with a modification of [ 10 ] to configurations admitting complex conjugated points due to E. Shustin (see [13]), I. Itenberg, V. Kharlamov and E. Shustin have been able to compute the polynomials χ 4 (T ) and χ 5 (T ), see [6].”
On Block-Goettsche multiplicities for planar tropical curves (2012, methodology): “For such a configuration X ⊂ R(2), the lattice paths algorithm [14] provides a bijection between certain subsets of integer points of ∆∗ and the set of δ-curves of genus g and degree ∆ which pass through the points of X .”
Welschinger invariant and enumeration of real rational curves (2003, background): “…point D4 and D(D +KΣ)/2− 2 nodes, (iv) reduced reducible curves, splitting into two rational nodal curves, intersecting transversely and only at their nonsingular points, (v) irreducible rational nodal curves with D(D+KΣ)/2+1 nodes such that exactly one of the nodes coincides with a point of w′.”
The tropical Grassmannian (2003, background): “Tropical curves are the key ingredient in Mikhalkin’s formula [9] for planar Gromov– Witten invariants.”
Logarithmic equivalence of Welschinger and Gromov-Witten invariants (2004, background): “It is taken from [5, 7] and its summary is found in [3].”
Exploded Manifolds (2009, methodology): “In this case, the number of holomorphic curves in the above exploded manifold passing through a given number of points can be calculated using Mikhalkin’s tropical methods from [9], by choosing the points to have tropical parts that appear in the interior of the above triangle in ‘tropical general p”
Tropical convexity (2003, background): “Our objective is to provide a combinatorial approach to convexity in the tropical semiring which is consistent with the recent developments in tropical algebraic geometry (see [15], [18], [20]).”