Monodromy and vanishing cycles for complete intersection curves

TL;DR

This paper computes the topological monodromy of families of complete intersection curves, revealing its relation to the r-spin mapping class group, and introduces new tools for analyzing monodromy via tensor product sections.

Contribution

It introduces a novel suite of tools for studying monodromy of sections of tensor products of line bundles, enabling inductive analysis of complete intersection curves.

Findings

Monodromy is given by the r-spin mapping class group.

Tools relate monodromy of tensor products to factors.

Method applies inductively to multi-degree cases.

Abstract

We compute the topological monodromy of every family of complete intersection curves. Like in the case of plane curves previously treated by the second-named author, we find the answer is given by the $r$-spin mapping class group associated to the maximal root of the adjoint line bundle. Our main innovation is a suite of tools for studying the monodromy of sections of a tensor product of very ample line bundles in terms of the monodromy of sections of the factors, allowing for an induction on (multi-)degree.