# Multigraded Factorial Rings and Fano varieties with torus action

@article{Hausen2009MultigradedFR, title={Multigraded Factorial Rings and Fano varieties with torus action}, author={Juergen Hausen and Elaine Herppich and Hendrik Suss}, journal={arXiv: Algebraic Geometry}, year={2009} }

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…

## 26 Citations

### ON TORUS ACTIONS OF HIGHER COMPLEXITY

- MathematicsForum of Mathematics, Sigma
- 2019

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

- MathematicsProceedings 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…

### NORMAL SINGULARITIES WITH TORUS ACTIONS

- Mathematics
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2017

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…

### COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

- MathematicsGlasgow Mathematical Journal
- 2015

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

- Mathematics
- 2017

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

- Mathematics
- 2015

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.…

## References

SHOWING 1-10 OF 38 REFERENCES

### Cox Rings and Combinatorics II

- Mathematics
- 2008

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

- Mathematics
- 2011

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…

### Canonical divisors on T-varieties

- Mathematics
- 2008

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

- Mathematics
- 2006

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

- Mathematics
- 2010

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

- Mathematics
- 1989

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

- Mathematics
- 1982

$k$ . $X$ 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

- Mathematics
- 2003

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…