Whitney equisingularity in families of generically reduced curves
Abstract
In this work we study equisingularity in a one-parameter flat family of generically reduced curves. We consider some equisingular criteria as topological triviality, Whitney equisingularity and strong simultaneous resolution. In this context, we prove that Whitney equisingularity is equivalent to strong simultaneous resolution and it is also equivalent to the constancy of the Milnor number and the multiplicity of the fibers. These results are extensions to the case of flat deformations of generically reduced curves, of known results on reduced curves. When the family is topologically trivial, we also characterize Whitney equisingularity through Cohen-Macaulay property of a certain local ring associated to the parameter space of the family.
Keywords
Cite
@article{arxiv.1809.03630,
title = {Whitney equisingularity in families of generically reduced curves},
author = {O. N. Silva and J. Snoussi},
journal= {arXiv preprint arXiv:1809.03630},
year = {2019}
}
Comments
Changes: A new result showing the connectivity of the fibers under the hypothesis on the constancy of the multiplicity of the fibers has been added (Lemma 4.6). Hence, the hypothesis about the connectivity of the fibers $X_t$ in Theorem 3.3 (first version) was omitted. Typos have been corrected, precise or missing references have been added. Some points that were not clear were better explained