Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

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.