Cross-ratio degrees and perfect matchings

TL;DR

This paper investigates cross-ratio degrees, which count point configurations satisfying constraints, and establishes an upper bound using perfect matchings on bipartite graphs, connecting to algebraic and tropical geometry.

Contribution

It introduces a new upper bound on cross-ratio degrees based on perfect matchings, linking combinatorics with algebraic geometry.

Findings

Established an upper bound on cross-ratio degrees

Connected cross-ratio degrees to perfect matchings in bipartite graphs

Provided multiple perspectives and example computations

Abstract

Cross-ratio degrees count configurations of points $z_1,\ldots, z_n \in \mathbb{P}^1$ satisfying $n - 3$ cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be viewed as combinatorial invariants of certain hypergraphs. We prove an upper bound on cross-ratio degrees in terms of the theory of perfect matchings on bipartite graphs. We also discuss several of the many perspectives on cross-ratio degrees -- including a connection to Gromov-Witten theory -- and give many example computations.