Labeled floor diagrams for plane curves

Sergey Fomin ; Grigory Mikhalkin

Journal of the European Mathematical Society, Tome 12 (2010) no. 6, pp. 1453-1496.

Résumé

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate rational curves of given degree passing through a collection of points on the complex plane and having maximal tangency to a given line. Another application of the combinatorial approach is a proof of a conjecture by P. Di Francesco–C. Itzykson and L. Göttsche that in the case of a fixed cogenus, the number of plane curves of degree d passing through suitably many generic points is given by a polynomial in d, assuming that d is sufficiently large. Furthermore, the proof provides a method for computing these “node polynomials”. A labeled floor diagram is obtained by labeling the vertices of a floor diagram by the integers 1; . . . ; d in a manner compatible with the orientation.We show that labeled floor diagrams of genus 0 are equinumerous to labeled trees, and therefore counted by the celebrated Cayley formula. The corresponding bijections lead to interpretations of the Kontsevich numbers (the genus-0 Gromov– Witten invariants of the projective plane) in terms of certain statistics on trees.

Détail

DOI : 10.4171/jems/238

Classification : 14-XX, 05-XX, 00-XX
Mots-clés : Gromov–Witten invariant, tropical curve, floor diagram, labeled tree