This is the first chapter of an introductory text under construction; further chapters are available via the authors' web pages Our aim is to provide an elementary access to Cox rings and their applications in algebraic and arithmetic geometry Any comments and suggestions on this draft will be highly appreciated

Cox Rings

https://www.research.ed.ac.uk/portal/files/16949862/The_Cox_ring_of_an_algebraic_variety_with_torus_action.pdf

The Cox ring of an algebraic variety with torus action

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 $ \mathbb{T} $ of dimension n. Let also ∂ be a homogeneous locally nilpotent derivation on the normal affine $ {\mathbb{Z}^n} $-graded domain A, so that ∂ generates a k+-action on X that is normalized by the $ \mathbb{T} $-action.

Affine t-varieties of complexity one and locally nilpotent derivations

Using the language of polyhedral divisors and divisorial fans we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one dimensional faces of it. Furthermore we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(D_h) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y.

Torus invariant divisors

Using the language of Altmann, Hausen and Sus, we describe invariant divisors on normal varieties X which admit an effective codimension one torus action. In this picture, X is given by a divisorial fan on a smooth projective curve Y. Cartier divisors on X can be described by piecewise affine functions h on the divisorial fan S whereas Weil divisors correspond to certain zero and one-dimensional faces of it. Furthermore, we provide descriptions of the divisor class group and the canonical divisor. Global sections of line bundles O(Dh) will be determined by a subset of a weight polytope associated to h, and global sections of specific line bundles on the underlying curve Y.

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 $ \mathbb{C} $
 *-surfaces and projectivizations of (nonsplit) vector bundles over toric varieties.

Gluing Affine Torus Actions Via Divisorial Fans

On terminal Fano 3-folds with 2-torus action

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. An important tool for the proof is a universal reduction of an arbitrary toric variety to a divisorial one. This is done in terms of support maps, a notion generalizing support functions on a polytopal fan. A further essential step is the decomposition of a given subtorus invariant regular map to a divisorial variety into an invariant toric part followed by a non-toric part.

/pdf/quotients-of-divisorial-toric-varities-51hygys3tv.pdf

Quotients of divisorial toric varities

A Non-Separated Version of Kajiwara's Construction

We associate to every divisorial (e.g. smooth) variety X with only constant invertible global functions and finitely generated Picard group a Pic(X)-graded homogeneous coordinate ring. This generalizes the usual homogeneous coordinate ring of the projective space and constructions of Cox and Kajiwara for smooth and divisorial toric varieties. We show that the homogeneous coordinate ring defines in fact a fully faithful functor. For normal complex varieties X with only constant global functions, we even obtain an equivalence of categories. Finally, the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated Picard group turns out to be a Krull ring admitting unique factorization.

/pdf/homogeneous-coordinates-for-algebraic-varieties-revised-2ha3lgxamb.pdf

Jürgen Hausen

Papers

Gluing Affine Torus Actions Via Divisorial Fans

On terminal Fano 3-folds with 2-torus action

Quotients of divisorial toric varities

A Non-Separated Version of Kajiwara's Construction

Homogeneous coordinates for algebraic varieties | Revised Version |