Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

Christoph Goldner

doi:10.37236/9422

Abstract

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio. This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet. We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

Highlights

We show that our recursive general Kontsevich’s formula implies the original Kontsevich’s formula and that the initial values are the numbers Kontsevich’s fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities
Consider the following enumerative problem: Determine the number Nd of rational degree d curves in P2C passing through 3d − 1 points in general position
We obtain a general tropical Kontsevich’s formula (Theorem 68) that recursively calculates the weighted number of rational plane tropical curves of degree d that satisfy point conditions, curve conditions and tropical cross-ratio conditions

Summary

Introduction

Consider the following enumerative problem: Determine the number Nd of rational degree d curves in P2C passing through 3d − 1 points in general position. The following question naturally comes up: Is there a general version of Kontsevich’s formula that recursively calculates the number of rational plane degree d curves that satisfy general positioned point, curve and cross-ratio conditions? Tyomkin’s correspondence theorem states that the number of rational plane degree d curves satisfying point and cross-ratio conditions equals its tropical counterpart. A general tropical Kontsevich’s formula that recursively computes the weighted number of rational plane tropical curves of degree d that satisfy point and tropical cross-ratio conditions simultaneously computes the classical numbers as well. We obtain a general tropical Kontsevich’s formula (Theorem 68) that recursively calculates the weighted number of rational plane tropical curves of degree d that satisfy point conditions, curve conditions and tropical cross-ratio conditions. We conclude the tropical and the classical Kontsevich’s formula from our general version

Preliminaries

Splitting curves with cross-ratios

Multiplicities of split curves

General Kontsevich’s formula