Tropical geometry and correspondence theorems via toric stacks

TL;DR

This paper extends correspondence theorems linking algebraic and tropical curves using toric stacks, providing a canonical tropicalization process and algebra-geometric proofs applicable in positive characteristic.

Contribution

It generalizes existing tropical correspondence theorems to include toric stacks and elliptic curves with fixed invariants, with a new algebraic proof framework.

Findings

Established a one-to-one correspondence between algebraic and tropical curves with toric constraints.

Introduced a canonical tropicalization procedure inspired by Berkovich's skeletons.

Proved theorems hold in large positive characteristic.

Abstract

In this paper we generalize correspondence theorems of Mikhalkin and Nishinou-Siebert providing a correspondence between algebraic and parameterized tropical curves. We also give a description of a canonical tropicalization procedure for algebraic curves motivated by Berkovich's construction of skeletons of analytic curves. Under certain assumptions, we construct a one-to-one correspondence between algebraic curves satisfying toric constraints and certain combinatorially defined objects, called "stacky tropical reductions", that can be enumerated in terms of tropical curves satisfying linear constraints. Similarly, we construct a one-to-one correspondence between elliptic curves with fixed $j$-invariant satisfying toric constraints and "stacky tropical reductions" that can be enumerated in terms of tropical elliptic curves with fixed tropical $j$-invariant satisfying linear constraints. Our theorems generalize previously published correspondence theorems in tropical geometry, and our proofs are algebra-geometric. In particular, the theorems hold in large positive characteristic.