Counting tropical rational curves with cross-ratio constraints

TL;DR

This paper develops tropical and combinatorial methods to count rational curves in toric surfaces with point and cross-ratio constraints, extending previous correspondence theorems and introducing new algorithms and diagrams.

Contribution

It introduces a lattice path algorithm and cross-ratio floor diagrams for enumerating rational curves with cross-ratio constraints in toric surfaces.

Findings

Established a tropical-algebraic correspondence for these counts.

Developed an explicit algorithm for tropical curve enumeration.

Introduced cross-ratio floor diagrams for simplified combinatorial analysis.

Abstract

We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces all tropical curves satisfying such degenerated conditions explicitly. In a special case simpler combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine these enumerative numbers as well.