Monodromy and vanishing cycles in toric surfaces

TL;DR

This paper characterizes which simple closed curves are vanishing cycles in nodal degenerations of smooth curves on toric surfaces, using monodromy group analysis to provide a complete classification.

Contribution

It offers a complete characterization of vanishing cycles for nodal degenerations on toric surfaces by analyzing the monodromy group associated with the linear system.

Findings

Complete description of vanishing cycles in toric surfaces

Determination of the monodromy group for the linear system

Reformulation of the problem via mapping class group monodromy

Abstract

Given an ample line bundle on a toric surface, a question of Donaldson asks which simple closed curves can be vanishing cycles for nodal degenerations of smooth curves in the complete linear system. This paper provides a complete answer. This is accomplished by reformulating the problem in terms of the mapping class group-valued monodromy of the linear system, and giving a precise determination of this monodromy group.