Triangulations and Maximal Cross-Ratio Degrees

TL;DR

This paper investigates the cross-ratio degree problem through tropical geometry, providing new formulas, solutions, and computational tools to count rational curves with specified conditions, extending known results up to n=9.

Contribution

It offers a tropical approach to compute cross-ratio degrees for triangulations, providing concrete solutions and a recursive algorithm that extends known counts to n=9.

Findings

Computed cross-ratio degrees for n=9 using tropical methods.

Provided concrete solutions for the counting problem in arbitrary settings.

Extended the known range of cross-ratio degrees from n=8 to n=9.

Abstract

The cross-ratio degree problem is about counting rational curves with $n$ marked points satisfying $n-3$ cross-ratio conditions. This problem has a tropical analogue which provides the same number, as shown by a correspondence theorem. In general, there are no closed formulas for this counting problem. In the special case of cross-ratio conditions given by triangulations, a formula was found by Silversmith via techniques of algebraic geometry. We study the cross-ratio problem given by triangulations in the tropical world. In addition to computing the cross-ratio degree by tropical means, we provide concrete solutions for the counting problem in arbitrary settings, thus answering the question by Silversmith. We also use the tropical recursive algorithm by Goldner to compute cross-ratio degrees to provide a new computational tool to compute cross-ratio degrees. With this, we can find all possible cross-ratio degrees for $n=9.$ Previously, these numbers were only known up to $n=8.$