Pencils and critical locus on normal surfaces

TL;DR

This paper investigates linear pencils of curves on normal surface singularities, linking the critical locus to special elements and providing a topological decomposition via minimal good resolution.

Contribution

It introduces a method to describe the topological type of generic and special elements of the pencil and relates the critical locus to these elements using minimal good resolution.

Findings

Critical locus is linked to special elements of the pencil.

Decomposition of the critical locus through minimal good resolution.

Provides topological information on the critical locus.

Abstract

We study linear pencils of curves on normal surface singularities. Using the minimal good resolution of the pencil, we describe the topological type of generic elements of the pencil and characterize the behaviour of special elements. Then we show that the critical locus associated to the pencil is linked to the special elements. This gives a decomposition of the critical locus through the minimal good resolution and as a consequence, information on the topological type of the critical locus.