We prove that if $\{f_t\}$ is a family of line singularities with constantL\^e numbers and such that $f_0$ is a homogeneous polynomial, then $\{f_t\}$ isequimultiple. 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 anapplication, 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 partialpositive answer to the famous Zariski multiplicity conjecture for a specialclass of non-isolated hypersurface singularities.
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机译：我们证明，如果$ \ {f_t \} $是具有常数L \ ^ e的线奇点族，并且$ f_0 $是齐次多项式，则$ \ {f_t \} $等倍。这将线奇点扩展为关于孤立奇点的A. M. Gabri \'elov和A. G. Ku \ v {s} nirenko的一个众所周知的定理。作为一个应用，我们表明如果$ \ {f_t \} $是拓扑上的$ \ mathscr {V} $等价的线奇点族，且$ f_0 $是同质的，则$ \ {f_t \} $等倍。这为非隔离超表面奇异点的特殊类提供了对著名的Zariski多重猜想的新的部分肯定答案。
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