Comptes Rendus
no. 8
Algebraic Geometry
Counting curves via lattice paths in polygons
[Calcul de courbes holomorphes par chemins en polygones]

Présenté par : Vladimir Arnold

Grigory Mikhalkin1, 2
1 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
2 St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia
Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 629-634.

Cette Note présente une formule pour les invariants énumératifs de genre arbitraire en surfaces toriques. La formule calcule le nombre des courbes de genre donné qui passent par une collection de points génériques sur la surface. Le résultat est donné en fonction de certains chemins dans le polygone de Newton relevant. Si la surface torique est ℙ 2 ou ℙ 1 ×ℙ 1 nos invariants sont les invariants de Gromov–Witten. La formule est nouvelle même dans ces cas, où d'autres techniques de calcul sont disponibles.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00104-3
Grigory Mikhalkin 1, 2

1 Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
2 St. Petersburg Branch of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, 191011, Russia 
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Grigory Mikhalkin. Counting curves via lattice paths in polygons. Comptes Rendus. Mathématique, Volume 336 (2003) no. 8, pp. 629-634. doi : 10.1016/S1631-073X(03)00104-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00104-3/

[1] L. Caporaso; J. Harris Counting plane curves of any genus, Invent. Math., Volume 131 (1998), pp. 345-392

[2] I.M. Gelfand; M.M. Kapranov; A.V. Zelevinski Discriminants, Resultants and Multidimensional Determinants, Birkhäuser, Boston, 1994

[3] M. Kontsevich; Yu. Manin Gromov–Witten classes, quantum cohomology and enumerative geometry, Comm. Math. Phys., Volume 164 (1994), pp. 525-562

[4] M. Kontsevich, Ya. Soibelman, Homological mirror symmetry and torus fibrations, | arXiv

[5] G. Mikhalkin, Gromov–Witten invariants and tropical algebraic geometry, to appear

[6] B. Sturmfels Solving Systems of Polynomial Equations, CBMS Regional Conf. Ser. in Math., American Mathematical Society, Providence, RI, 2002

[7] R. Vakil Counting curves on rational surfaces, Manuscripta Math., Volume 102 (2000), pp. 53-84

[8] J.-Y. Welschinger, Invariants of real rational symplectic 4-manifolds and lower bounds in real enumerative geometry, C. R. Acad. Sci. Paris, 2003, to appear

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