• Corpus ID: 13911682

Multigraded Factorial Rings and Fano varieties with torus action

  title={Multigraded Factorial Rings and Fano varieties with torus action},
  author={Juergen Hausen and Elaine Herppich and Hendrik Suss},
  journal={arXiv: Algebraic Geometry},
In a first result, we describe all finitely generated factorial algebras over an algebraically closed field of characteristic zero that come with an effective multigrading of complexity one by means of generators and relations. This enables us to construct systematically varieties with free divisor class group and a complexity one torus action via their Cox rings. For the Fano varieties of this type that have a free divisor class group of rank one, we provide explicit bounds for the number of… 


We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms

On Fano Varieties with Torus Action of Complexity 1

  • E. Herppich
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 2014
Abstract In this work we provide effective bounds and classification results for rational ℚ-factorial Fano varieties with a complexity-one torus action and Picard number 1 depending on the two


We propose a method to compute a desingularization of a normal affine varietyX endowed with a torus action in terms of a combinatorial description of such a variety due to Altmann and Hausen. This

Factorially graded rings of complexity one

We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the

Classifying Fano complexity-one T-varieties via divisorial polytopes

The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one T-varieties and


Abstract We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring,

Beyond Toric Geometry

Over the past several decades, toric geometry has become an increasingly important area of algebraic geometry. On the one hand, many deep theorems in algebraic geometry can be reduced to statements

Non‐complete rational T‐varieties of complexity one

We consider rational varieties with a torus action of complexity one and extend the combinatorial approach via the Cox ring developed for the complete case in earlier work to the non‐complete, e.g.

Divisor class groups of rational trinomial varieties



Cox Rings and Combinatorics II

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

Torus invariant divisors

Using the language of Altmann, Hausen and Süß, 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

Homogeneous coordinates for algebraic varieties

Canonical divisors on T-varieties

Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their

Gluing Affine Torus Actions Via Divisorial Fans

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

Cohomological and geometric approaches to rationality problems: New perspectives

Preface.- Unremified cohomology of finite groups of Lie type.- The rationality of the moduli space of curves of genus 3 after P. Katsylo.- The rationality of certain moduli spaces of curves of genus

The Picard Group of a G-Variety

Let G be a reductive algebraic group and X an algebraic G-variety which admits a quotient it: X → X//G. In this article we describe several results concerning the Picard group Pic(X//G) of the

The Classification of Fano 3-Folds with Torus Embeddings

$k$ . $X$ is called a Fano 3-fold if the anti-canonical divisor $-K_{X}$ of $X$ is ample. Recently, Ishkovsky has developped the theory of Fano 3-folds in his papers [1], [2] and has determined the

Polyhedral divisors and algebraic torus actions

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