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.

In this paper we classify SL2-actions on normal affine T-varieties that are normalized by the torus T. This is done in terms of a combinatorial description of T-varieties given by Altmann and Hausen. The main ingredient is a generalization of Demazure's roots of the fan of a toric variety. As an application we give a description of special SL2-actions on normal affine varieties. We also obtain, in our terms, the classification of quasihomogeneous SL2-threefolds due to Popov. IntroductionLet k be an algebraically closed field of characteristic zero, M be a lattice of rank n, N = Hom(M, Z) be the dual lattice of M , and T be the algebraic torus Spec k[M ], so that M is the character lattice of T and N is the one-parameter subgroup lattice of T.A T-variety X is a normal algebraic variety endowed with an effective regular action of T. The complexity of a T-action is the codimension of a general orbit, and since the T-action on X is effective, the complexity of X equals dim X − rank M . For an affine variety X, to introduce a T-action on X is the same as to endow k[X] with an M -grading. There are well known combinatorial descriptions of T-varieties. We send the reader to [Dem70] and [Ful93] for the case of toric varieties, to [KKMS73, Ch. 2 and 4] and [Tim08] for the complexity one case, and to [AH06, AHS08] for the general case. In this paper we use the approach in [AH06].Any affine toric variety is completely determined by a polyhedral cone σ ⊆ N Q . Similarly, the description of a normal affine T-varieties X due to Altmann and Hausen [AH06] involves the data (Y, σ, D) where Y is a normal semiprojective variety, σ ⊆ N Q := N ⊗ Q is a polyhedral cone, and D is a divisor on Y whose coefficients are polyhedra in N Q with tail cone σ. The divisor D is called a σ-polyhedral divisor on Y (see Section 1.1 for details).Let X be a T-variety endowed with a regular G-action, where G is any linear algebraic group. We say that the G-action on X is compatible if the image of G in Aut(X) is normalized but not centralized by T. Furthermore, we say that the G-action is of fiber type if the general orbits are contained in the T-orbit closures, and of horizontal type otherwise [FZ05, Lie10a].Let now G a = G a (k) be the additive group of k. It is well known that a G a -action on an affine variety X is equivalent to a locally nilpotent derivation (LND) of k[X]. A description of compatible G a -actions on an affine T-variety, or equivalently of homogeneous LNDs on k[X], is available in the case where X is of complexity at most one [Lie10a] or the G a -action is of fiber type [Lie10b] in terms of a generalization of Demazure's roots of a fan [Dem70] (see Sections 1.3 and 1.4).A regular SL 2 -action on an affine variety X is uniquely defined by an sl 2 -triple {δ, ∂ + , ∂ − } of derivations of the algebra k[X], where ∂ ± are locally nilpotent, δ = [∂ + , ∂ − ] is semisimple and [δ, ∂ ± ] = ±2∂ ± (see Proposition 2.1). Assume now that X is an affine T-variety. If the

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Anders\'en-Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X-Y is different from T. For toric surfaces we are able to classify those which posses a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.Comment: 15 pages. Minor corrections. To appear in Journal of Pure and Applied Algebr

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Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T = G n m stands for the split algebraic torus. Our main result is a classification of additive group actions on X that are normalized by the T-action. This generalizes the classification given by the second author in the particular case where k is algebraically closed and of characteristic zero. With the assumption that the characteristic of k is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine T-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these G a -actions in the toric situation.