The vanishing cycles of curves in toric surfaces I

TL;DR

This paper investigates the conditions under which simple closed curves in curves on toric surfaces can be contracted as vanishing cycles, using tropical geometry to identify when such cycles exist.

Contribution

It provides a list of obstructions to contracting vanishing cycles and demonstrates that non-separating simple closed curves are vanishing cycles when obstructions are absent.

Findings

Identifies obstructions to contracting vanishing cycles in toric surfaces.

Shows non-separating simple closed curves are vanishing cycles absent obstructions.

Utilizes tropical geometry to analyze vanishing cycles.

Abstract

This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical means, we show that any non-separating simple closed curve is a vanishing cycle whenever none of the listed obstructions appears.