Zariski's Multiplicity Question --- A Survey - NZJM

New Zealand Journal of Mathematics

Christophe Eyral¹ 

[1] Max-Planck Institut für MathematikVivatsgasse 753111 BonnGERMANY$$

关键词: Complex hypersurface singularity;    embedded topological type;    multiplicity.;   

DOI  :  

学科分类：社会科学、人文和艺术（综合）

来源: University of Auckland * Department of Mathematics

【 摘 要 】

In 1971, Zariski asked the following question: is the multiplicity of a reduced analytic hypersurface singularity in depends only on its embedded topological type? More precisely, if are reduced germs (at the origin) of holomorphic functions such that the corresponding germs of hypersurfaces in have the same embedded topological type, then is it true that f and g have the same multiplicity at? Instead of dealing with a pair (f,g), one can also consider a similar question for a family (Ft)t. More precisely, if is a reduced germ of holomorphic function and (Ft)t a topologically -constant (holomorphic) deformation of it, then is it true that (Ft)t is equimultiple? Almost 40 years later, these problems are, in general, still unsettled (even for hypersurfaces with isolated singularities). The answer to the first question is, nevertheless, known to be positive in the special case of plane curve singularities (i.e., when n = 2). Concerning families, equimultiplicity is known for topologically -constant deformations of semiquasihomogeneous and quasihomogeneous isolated singularities and topologically -constant deformations within a family of convenient Newton nondegenerate isolated singularities. Equimultiplicity is also known for topologically -constant deformations within some very special families of nonisolated singularities. Moreover, it is known that the multiplicity is an embedded C1 invariant as well as an embedded topological right-left bilipschitz invariant. Several other (more specific) results are also known and will be mentioned in this survey. Our aim here is to provide in a short exposition a general overview of Zariski's problem which is one of the most fascinating (but also very difficult!) problem in equisingularity theory.

【 授权许可】

Unknown