We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.

We investigate the Cox ring of a normal complete variety X with algebraic torus action. Our first results relate the Cox ring of X to that of a maximal geometric quotient of X. As a consequence, we obtain a complete description of the Cox ring in terms of generators and relations for varieties with torus action of complexity one. Moreover, we provide a combinatorial approach to the Cox ring using the language of polyhedral divisors. Applied to smooth K * -surfaces, our results give a description of the Cox ring in terms of Orlik-Wagreich graphs. As examples, we explicitly compute the Cox rings of all Gorenstein del Pezzo K * -surfaces with Picard number at most two and the Cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of Klyachko's description.

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combinatorially minimal one.Remark 2.1. Let s : Cl(X) → D be any set-theoretical section of the canonical map D → Cl(X). Then the projection S → R restricts to a canonical isomorphism of sheaves of Cl(X)-graded vector spaces:One can show that, up to isomorphism, the graded O X -algebra R does not depend on the choices of D and the shifting family, compare [3, Lemma 3.7]. We define the Cox ring R(X) of X, also called the total coordinate ring of X, to be the Cl(X)-graded algebra of global sections of the Cox sheaf: R(X) := Γ(X, R) ∼ = Γ(X, S)/Γ(X, I).The sheaf R defines a universal torsor p : X → X in the following sense. Suppose that R is locally of finite type; this holds for example, if X is locally factorial or if R(X) is finitely generated. Then we may consider the relative spectrum X := Spec X (R).The Cl(X)-grading of the sheaf R defines an action of the diagonalizable group H := Spec(K[Cl(X)]) on X, and the canonical morphism p : X → X is a good quotient, i.e., it is an H-invariant affine morphism satisfying

Generalizing the passage from a fan to a toric variety, we provide a combinatorial approach to construct arbitrary effective torus actions on normal, algebraic varieties. Based on the notion of a "proper polyhedral divisor" introduced in earlier work, we develop the concept of a "divisorial fan" and show that these objects encode the equivariant gluing of affine varieties with torus action. We characterize separateness and completeness of the resulting varieties in terms of divisorial fans, and we study examples like C * -surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

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