Equinormalizable theory for plane curve singularities with embedded points and the theory of equisingularity

Abstract

In this paper we give some criteria for a family of generically reduced plane curve singularities to be equinormalizable. The first criterion is based on the $\delta$-invariant of a (non-reduced) curve singularity which is introduced by Brücker-Greuel (\cite{BG}). The second criterion is based on the I-equisingularity of a $k$-parametric family ($k\ge 1$) of generically reduced plane curve singularities, which is introduced by Nobile (\cite{No}) for one-parametric families. The equivalence of some kinds of equisingularities of a family of generically reduced plane curve singularities is also studied.

16 pages, replaced Example 2.1 and Example 3.1, added Example 4.1 and Example 4.2, to appear in Hokkaido Mathematical Journal 2011