Labeled floor diagrams for plane curves

TL;DR

This paper introduces labeled floor diagrams as a combinatorial tool to compute and interpret enumerative invariants of plane curves, providing explicit formulas and proofs for polynomiality of node counts.

Contribution

It defines labeled floor diagrams for plane curves, establishes their enumeration, and connects them to Gromov-Witten invariants and tree structures, offering new combinatorial methods.

Findings

Labeled floor diagrams of genus 0 correspond to labeled trees, counted by Cayley's formula.

The approach yields explicit formulas for enumerating rational curves with specified tangency.

Proof of polynomiality of node counts for fixed cogenus in large degree cases.

Abstract

Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle 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. Goettsche 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's 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.