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.
Categories: math.AG math.CV