Correspondence theorems for tropical curves I

TL;DR

This paper develops a new method to analyze the deformation of degenerate algebraic curves on singular varieties, providing criteria for when such curves can be smoothed, with applications to genus one curves on toric degenerations.

Contribution

It introduces a systematic approach to compute obstruction classes for degenerate curves, extending realizability results for tropical genus one curves.

Findings

Derived necessary and sufficient conditions for tropical genus one curve realizability.

Developed a new method for calculating obstruction cohomology classes.

Applied the method to toric degenerations of algebraic varieties.

Abstract

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the obstruction cohomology class of degenerate algebraic curves. This enables us to judge whether a given degenerate curve can be deformed to a smooth curve or not in variety of situations. In this paper, we apply it to curves of genus one on degeneration of toric varieties. In particular, we obtain the necessary and sufficient condition for the realizability of tropical curves of genus one, extending various results obtained so far.