This is the first chapter of an introductory text under construction; further chapters are available on the author's web pages. Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry. We are grateful to Victor Batyrev and Alexei Skorobogatov for helpful remarks and discussions. Any comments and suggestions on this draft will be highly appreciated.

Abstract. Given an irreducible affine algebraic variety X of dimension n ≥ 2, we let SAut(X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x ∈ X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X). We provide also various modifications and applications.

Abstract. We say that a group G acts infinitely transitively on a set X if for every m ∈ N the induced diagonal action of G is transitive on the cartesian mth power X m \ ∆ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of affine cones over flag varieties, the second of non-degenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups of a reinforced type.

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions.As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.Theorem 2. Let X 1 be a rational, normal, affine variety with a torus action of complexity one of Type 2 and at most log terminal singularities. Then there is a unique chain of quotients

No abstract