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Affine T-varieties: additive group actions and singularities

Abstract : A T-variety is an algebraic variety endowed with an effective action of an algebraic torus T. This thesis is devoted to the study of two aspects of normal affine T-varieties: the additive group actions and the characterization of singularities. Let X = Spec A be a normal affine T-variety and let D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates an action of the additive group on X. We provide a complete classification of pairs (X, D) in three cases: for toric varieties, in the case where the complexity is one, and in the case where D is of fiber type. As an application, we compute the homogeneous Makar-Limanov (ML) invariant of such varieties. We deduce that any variety with trivial ML-invariant is birationally decomposable as Y × P^2, for some variety Y. Conversely, given a variety Y, there exists an affine T-variety X with trivial ML invariant birational to Y × P2. In the second part concerning singularities of a T-variety X we compute the higher direct images of the structure sheaf of a desingularization of X. As a consequence, we give a criterion as to when a T-variety has rational singularities. We also provide a condition for a T-variety to be Cohen-Macaulay. As an application, we characterize quasihomogeneous elliptic singularities of surfaces.
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Contributor : Alvaro Liendo <>
Submitted on : Wednesday, May 11, 2011 - 8:47:43 PM
Last modification on : Wednesday, November 4, 2020 - 2:41:49 PM
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  • HAL Id : tel-00592274, version 1



Alvaro Liendo. Affine T-varieties: additive group actions and singularities. Mathematics [math]. Université de Grenoble, 2010. English. ⟨tel-00592274⟩



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