Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces

TL;DR

This paper computes the monodromy of ample linear systems on simply connected surfaces, linking it to r-spin mapping class groups and characterizing vanishing cycles in degenerations, with implications for Lefschetz fibrations and 4-manifolds.

Contribution

It identifies the monodromy with r-spin mapping class groups and characterizes vanishing cycles for linear systems on simply connected surfaces.

Findings

Monodromy corresponds to r-spin mapping class groups.

Characterization of vanishing cycles in nodal degenerations.

Implications for discriminants, Lefschetz fibrations, and 4-manifolds.

Abstract

We compute the mapping class group-valued monodromy of any sufficiently ample linear system on any smooth simply connected projective surface, identifying this with the r-spin mapping class group associated to a maximal root of the adjoint line bundle. This gives a characterization of the simple closed curves that can arise as vanishing cycles for nodal degenerations in the linear system, as well as other corollaries concerning discriminants, Lefschetz fibrations, and surfaces in 4-manifolds.