Let X = Spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine Z n -graded domain A, so that ∂ generates a k + -action on X that is normalized by the T-action.We provide a complete classification of pairs (X, ∂) in two cases: for toric varieties (n = dim X) and in the case where n = dim X − 1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular, we exhibit a family of nonrational varieties with trivial Makar-Limanov invariant.
We propose a method to compute a desingularization of a normal affine variety X endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This desingularization allows us to study the structure of the singularities of X. In particular, we give criteria for X to have only rational, (Q-)factorial, or (Q-)Gorenstein singularities. We also give partial criteria for X to be Cohen-Macaulay or log-terminal.Finally, we provide a method to construct factorial affine varieties with a torus action. This leads to a full classification of such varieties in the case where the action is of complexity one.
We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo K * -surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number one having a reductive automorphism group with two-dimensional maximal torus.
Abstract. In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of P n+1 , for a fixed n ≥ 2. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p < 2 n+1 and that any such hypersurface admitting an automorphism of order p > 2 n is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli space of principally polarized abelian varieties.
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