Counting tropical rational curves with cross-ratio constraints

Christoph Goldner

doi:10.1007/s00209-020-02506-8

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 (Tyomkin in Adv Math 305:1356–1383, 2017), 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 rational 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.

Highlights

Tropical geometry is a rather young field of mathematics that is intimately connected to algebraic geometry, non-Archimedean analytic geometry and combinatorics
A correspondence theorem states that an enumerative number equals its tropical counterpart, where in tropical geometry we have to count each tropical object with a suitable multiplicity reflecting the number of classical objects in our counting problem that tropicalize to the given tropical object
Tropical geometry hands us a new approach to enumerative problems: first find a suitable correspondence theorem, use combinatorics to enumerate the tropical objects in question

Summary

Introduction

Tropical geometry is a rather young field of mathematics that is intimately connected to algebraic geometry, non-Archimedean analytic geometry and combinatorics. Tropical geometry offers a new approach to compute the numbers Nd , and generalizations thereof: in [22], Mikhalkin pioneered the use of tropical methods in enumerative geometry by proving a correspondence theorem for counts of curves in toric surfaces satisfying point conditions. The correspondence theorem we are going to use is provided by Tyomkin in [31] It describes the multiplicities with which parameterized rational tropical curves have to be counted when they satisfy cross-ratio constraints. The second step to answer questions (1) and (2) is to explicitly construct all parameterized rational tropical curves that satisfy the given point and degenerated cross-ratio conditions using combinatorial methods.

Combinatorial methods

Preliminaries

Tropical cross-ratios

Cross-ratio lattice path algorithm

Duality: tropical curves and subdivisions

Floor diagrams for cross-ratio counts