Vanishing arcs for isolated plane curve singularities

TL;DR

This paper introduces vanishing arcsets as a topological analogue of classical vanishing cycles for isolated plane curve singularities, characterizing their properties and connections to monodromy and A'Campo's divides.

Contribution

It develops a new topological framework using vanishing arcsets, linking geometric variation operators with intersection theory and existing vanishing cycle collections.

Findings

Characterization of arcs sent to vanishing cycles via intersection numbers.

Existence of arcset collections matching vanishing cycles from A'Campo's divides.

Extension of classical theory to a purely topological setting.

Abstract

The variation operator associated with an isolated hypersurface singularity is a classical topological invariant that relates relative and absolute homologies of the Milnor fiber via a non trivial isomorphism. Here we work with a topological version of this operator that deals with proper arcs and closed curves instead of homology cycles. Building on the classical framework of geometric vanishing cycles, we introduce the concept of vanishing arcsets as their counterpart using this geometric variation operator. We characterize which properly embedded arcs are sent to geometric vanishing cycles by the geometric variation operator in terms of intersections numbers of the arcs and their images by the geometric monodromy. Furthermore, we prove that for any distinguished collection of vanishing cycles arising from an A'Campo's divide, there exists a topological exceptional collection of arcsets whose variation images match this collection.