Whitney equisingularity in families of generically reduced curves
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O. N. Silva
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
· J. Snoussi
Whitney equisingularity in families of generically reduced curves Received: 10 September 2018 / Accepted: 8 November 2019 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.
1. Introduction Consider a germ of reduced equidimensional complex surface (X, 0) together with an analytic flat map p from (X, 0) to (C, 0) and consider a representative p : X → T (see Definition 2.1(c)). The surface X can be viewed as a one parameter flat deformation of the curve X 0 := p −1 (0). When (X, 0) is not Cohen–Macaulay the germ of curve (X 0 , 0) has an embedded component at the origin. If (X 0 , 0) is reduced at all its points x = 0, then we say that (X 0 , 0) is a germ of generically reduced curve. We can see that this situation arises in many and natural examples (see [6, Ex. 55], [7, Ex. 2.4], [8, Ex. 9.11], [10, Ex. 4.6] and [13, Prop. 3.51]). Brücker and Greuel studied these deformations in [2] and defined a δ invariant and a Milnor number for a generically reduced curve. In particular, they showed that normalization in family is equivalent to the constancy of δ of the fibers along the parameter space [2, Kor. 3.2.1]. In [10], Công–Trình Lê studied topological triviality of such families comparing it to the constancy of Milnor number; the full statement was proved by Greuel in [8, Th. 9.3]. In [7, Th. 3.1] Whitney equisingularity of such a family was proved to be equivalent to Zariski’s discriminant criterion. O. N. Silva (B)· J. Snoussi: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Unidad Cuernavaca, Av. Universidad s/n, Lomas de Chamilpa, 62210 Cuernavaca, Morelos, Mexico. e-mail: [email protected] J. Snoussi e-mail: [email protected] Mathematics Subject Classification: 32S15
https://doi.org/10.1007/s00229-019-01164-3
O. N. Silva, J. Snoussi
All these results are extensions of known results on reduced curves to the case of generically reduced curves. See [1] and [3] for more details on the reduced case. The aim of this work is to study Whitney equisingular deformations of a generically reduced curve. Our first main result is the characterization of Whitney equisingularity by the constancy of the Milnor number and the multiplicity of the fibers along the parameter space (Theorem 4.7). The second main res
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