General Kontsevich-style formula for Hirzebruch Surfaces

TL;DR

This paper extends Kontsevich-style formulas to count rational curves in Hirzebruch surfaces with point and cross-ratio conditions using tropical geometry, generalizing previous results and allowing more flexible conditions.

Contribution

It provides a new Kontsevich-style formula for Hirzebruch surfaces _r, incorporating multiple cross-ratio conditions with tropical methods, broadening the scope of curve counting.

Findings

Derived a formula for _r surfaces using tropical geometry.

Extended curve counting to include multiple cross-ratio conditions.

Demonstrated increased flexibility in imposing geometric conditions.

Abstract

Tyomkin's correspondence theorem states the equality of counts of rational curves of fixed homology class in a toric surface satisfying point and cross-ratio conditions with their tropical counterparts. Such correspondence theorems allow us to derive non-tropical results from tropical ones; for example, Mikhalkin's correspondence theorem is used in the tropical proof of the famous Kontsevich formula for counts of plane rational curves of degree $d$ satisfying point conditions. This formula has been generalized to counts of curves in the Hirzebruch surface $\mathbb{F}_{2}$ satisfying point conditions. Further generalizations allow curves in $\mathbb{P}^2$ to satisfy multiple cross-ratio conditions. In this paper, we present a Kontsevich-style formula for the Hirzebruch surface $\mathbb{F}_r$, $r \in \mathbb{N}$, which counts rational tropical curves of a fixed homology class satisfying point and multiple cross-ratio conditions using tropical methods. Moreover, the cross-ratio conditions we impose on the curves allow more freedom.