Additive group actions on affine T-varieties of complexity one in arbitrary characteristic

Let X be a normal affine T-variety of complexity at most one over a perfect field k, where T 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 additive group actions in the toric situation.Comment: 31 page

Automorphisms of prime order of smooth cubic n-folds

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 > 1. 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.Comment: 10 page

Roots of the affine Cremona group

Let k [n] = k[x 1, , x n ] be the polynomial algebra in n variables and let ${\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}}$ . In this note we show that the root vectors of ${\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right)$ , the subgroup of volume preserving automorphisms in the affine Cremona group ${\text{Aut}}\left( {{\mathbb{A}^n}} \right)$ , with respect to the diagonal torus are exactly the locally nilpotent derivations x α (∂/∂x i ), where x α is any monomial not depending on x i . This answers a question posed by Popo

Cohen-Macaulay Du Bois singularities with a torus action of complexity one

Using Altmann-Hausen-S\"u\ss\ description of T-varieties via divisorial fans and K\'ovacs-Schwede-Smith characterization of Du Bois singularities we prove that any rational T-variety of complexity one which is Cohen-Macaulay and Du Bois has rational singularities. In higher complexity, we prove an analogous result in the case where the Chow quotient of the T-variety has Picard rank one and trivial geometric genus.Comment: 16 page