Some Special Cases of Bobadilla's Conjecture

TL;DR

This paper proves two specific cases of Bobadilla's conjecture for hypersurfaces with 1-dimensional critical loci using a new algebraically computable invariant called the beta invariant, linking it to the conjecture's hypotheses.

Contribution

It introduces the beta invariant and demonstrates its effectiveness in proving special cases of Bobadilla's conjecture for certain hypersurfaces.

Findings

Two special cases of Bobadilla's conjecture are proved.

The beta invariant is shown to be a key tool in understanding the conjecture.

Vanishing of the beta invariant characterizes the conjecture's conditions.

Abstract

We prove two special cases of a conjecture of J. Fern\'andez de Bobadilla for hypersurfaces with $1$-dimensional critical loci. We do this via a new numerical invariant for such hypersurfaces, called the beta invariant, first defined and explored by the second author in 2014. The beta invariant is an algebraically calculable invariant of the local ambient topological-type of the hypersurface, and the vanishing of the beta invariant is equivalent to the hypotheses of Bobadilla's conjecture.