Enumeration of rational curves with cross-ratio constraints

TL;DR

This paper establishes an algebraic-tropical correspondence for rational curves with marked points satisfying cross-ratio constraints in toric varieties, generalizing previous results and applicable in all characteristics.

Contribution

It proves a new correspondence theorem for rational stable maps with cross-ratio conditions, extending Nishinou and Siebert's work to more general settings and arbitrary characteristics.

Findings

Proves algebraic-tropical correspondence for specified stable maps

Generalizes previous results to include boundary tangency and cross-ratio constraints

Applicable in arbitrary characteristic, including mixed characteristic

Abstract

In this paper we prove the algebraic-tropical correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus and in the boundary divisor, the map has prescribed tangency to the boundary divisor, and certain quadruples of marked points have prescribed cross-ratios. In particular, our results generalize the results of Nishinou and Siebert. The proof is very short, involves only the standard theory of schemes, and works in arbitrary characteristic (including the mixed characteristic case).