Title: Tropical geometry and correspondence theorems via toric stacks
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 ca...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.Read More
Publication Year: 2010
Publication Date: 2010-01-10
Language: en
Type: preprint
Access and Citation
Cited By Count: 5
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot