Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

TL;DR

This paper proves that families of line singularities with constant invariants and a homogeneous initial polynomial are equimultiple, extending classical results and providing partial evidence for the Zariski multiplicity conjecture in non-isolated cases.

Contribution

It extends known theorems to line singularities, showing that topologically trivial families with homogeneous initial conditions are equimultiple, advancing understanding of hypersurface singularities.

Findings

Families with constant L extsuperscript{} numbers are equimultiple.

Topologically }-equisingular families with homogeneous initial polynomial are equimultiple.

Provides partial positive evidence for the Zariski multiplicity conjecture.

Abstract

We prove that if $\{f_t\}$ is a family of line singularities with constant L\^e numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ is equimultiple. This extends to line singularities a well known theorem of A. M. Gabri\`elov and A. G. Ku\v{s}nirenko concerning isolated singularities. As an application, we show that if $\{f_t\}$ is a topologically $\mathscr{V}$-equisingular family of line singularities, with $f_0$ homogeneous, then $\{f_t\}$ is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.