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 equisingularity 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 (X, 0) 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.

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Acknowledgements

The authors warmly thank the referees for very careful reading and valuable comments and suggestions. We would like to thank M.A.S Ruas for many helpful conversations and G-M. Greuel for his suggestions and comments on this work. The first author would like to thank CONACyT for the financial support by Fordecyt 265667. Both authors are grateful to UNAM/DGAPA for support by PAPIIT IN 113817, and to CONACyT Grant 282937.

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Correspondence to O. N. Silva.

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Silva, O.N., Snoussi, J. Whitney equisingularity in families of generically reduced curves. manuscripta math. 163, 463–479 (2020). https://doi.org/10.1007/s00229-019-01164-3

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Mathematics Subject Classification

  • 32S15