Abstract
The aim of this note is to prove, in the spirit of a rigidity result for isolated singularities of Schlessinger see Schlessinger (Invent Math 14:17–26, 1971) or also Kleiman and Landolfi (Compositio Math 23:407–434, 1971), a variant of a rigidity criterion for arbitrary singularities (Theorem 2.1 below). The proof of this result does not use Schlessinger’s Deformation Theory [Schlessinger (Trans Am Math Soc 130:208–222, 1968) and Schlessinger (Invent Math 14:17–26, 1971)]. Instead it makes use of Local Grothendieck-Lefschetz Theory, see (Grothendieck 1968, Éxposé 9, Proposition 1.4, page 106) and a Lemma of Zariski, see (Zariski, Am J Math 87:507–536, 1965, Lemma 4, page 526). I hope that this proof, although works only in characteristic zero, might also have some interest in its own.
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Acknowledgements
I want to thank the referee for his useful remarks, for suggesting some improvements of the presentation, and for asking to include a proof of Lemma 1.6.
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To the memory of my teacher and dear friend Alexandru Lascu.
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Bădescu, L. On a rigidity result of singularities. Ann Univ Ferrara 63, 25–32 (2017). https://doi.org/10.1007/s11565-016-0267-6
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DOI: https://doi.org/10.1007/s11565-016-0267-6