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Transactions of the American Mathematical Society

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Equivariant $ G$-structure on versal deformations


Author: Dock S. Rim
Journal: Trans. Amer. Math. Soc. 257 (1980), 217-226
MSC: Primary 14D15
DOI: https://doi.org/10.1090/S0002-9947-1980-0549162-8
MathSciNet review: 549162
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Abstract: Let $ {X_0}$ be an algebraic variety, and $ (\chi ,\,\Sigma )$ its versal deformation. Now let G be an affine algebraic group acting algebraically on $ {X_0}$. It gives rise to a definite linear G-action on the tangent space of $ \Sigma $. In this paper we establish that if G is linearly reductive then there is an equivariant G-action on $ (\chi ,\Sigma )$ which induces given G-action on the special fibre $ {X_0}$ and its linear G-action on the tangent space of the formal moduli $ \Sigma $. Furthermore, such equivariant G-structure is shown to be unique up to noncanonical isomorphism.


References [Enhancements On Off] (What's this?)

  • [1] A. Grothendieck, Séminaire de géométrie algébrique. II, Exp. 6, Inst. Hautes Etudes Sci., Paris, 1960-1961.
  • [2] -, Techniques de construction et théorèmes d'existence en géométrie algébrique. IV: les schèmes de Hilbert, Seminaire Bourbaki, Exp. 221, Secrétariat Math., Paris, 1961.
  • [3] H. C. Pinkham, Deformations of cones with negative grading, J. Algebra 30 (1974), 92-102. MR 0347822 (50:323)
  • [4] -, Deformations of algebraic varieties with $ {G_m}$ action, Astérique No. 20, Soc. Math. France, Paris, 1974.
  • [5] Dock S. Rim, Formal moduli of deformation, Lecture Notes in Math., vol. 288, Springer-Verlag, Berlin and New York, 1972, pp. 32-132.
  • [6] -, Torsion differentials and deformation, Trans. Amer. Math. Soc. 169 (1972), 275-278. MR 49 #7259. MR 0342513 (49:7259)
  • [7] Dock S. Rim and M. Vitulli, Weierstrass points and monomial curves, J. Algebra 48 (1977), 454-476. MR 76 #15652. MR 0457447 (56:15652)
  • [8] M. Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208-222. MR 36 #184. MR 0217093 (36:184)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0549162-8
Keywords: Infinitesimal deformation, formal moduli, versal deformation, linearly reductive group, fibred category in groupoid, cohomology of linearly reductive group
Article copyright: © Copyright 1980 American Mathematical Society

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