Graphs and obstruction theory for algebraic curves

TL;DR

This paper explores the relationship between algebraic and tropical curves using obstruction theory, emphasizing higher valent vertices, and provides new formulas, criteria, and correspondence results for their study.

Contribution

It introduces a general formula for obstructions, a new vanishing criterion, and extends the correspondence between algebraic and tropical curves to regular tropical curves.

Findings

Derived a formula describing obstructions in algebraic curve deformations.

Established a criterion for the vanishing of obstructions.

Proved an optimal correspondence between tropical and algebraic curves for regular tropical curves.

Abstract

In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs can be encoded in so-called tropical curves. Our main concern is the relation between algebraic curves and tropical curves where the deformation problem is obstructed. Particular emphasis is put on the role of higher valent vertices of tropical curves, which has not been developed well so far in spite of its importance in this area of study. We will give a general formula describing the obstruction, a new criterion for the vanishing of the obstruction, and a relation between the number of algebraic curves and the number of integral points in certain polytopes. We also prove the optimal version of the correspondence between tropical curves and algebraic curves when the tropical curves are regular, generalizing previous results.