# Non‐complete rational T‐varieties of complexity one

@article{Hausen2015NoncompleteRT, title={Non‐complete rational T‐varieties of complexity one}, author={Juergen Hausen and Milena Wrobel}, journal={Mathematische Nachrichten}, year={2015}, volume={290} }

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. affine, case. This includes in particular a description of all factorially graded affine algebras of complexity one with only constant homogeneous invertible elements in terms of canonical generators and relations.

## 23 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…

Limit points and additive group actions

- MathematicsRicerche di Matematica
- 2021

We show that an effective action of the one-dimensional torus Gm on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product Gm ⋌ Ga with the same general…

Crepant Resolutions of 3-Dimensional Quotient Singularities via Cox Rings

- MathematicsExp. Math.
- 2019

Cox rings of crepant resolutions of quotient singularities are studied to obtain information on the geometric structure of these resolutions, number of different resolutions, and relations between them.

On rigidity of factorial trinomial hypersurfaces

- MathematicsInt. J. Algebra Comput.
- 2016

It is proved that a factorial trinomial hypersurface is rigid if and only if every exponent in the trin coefficients is at least 2.

On deformations of toric varieties

- Mathematics
- 2016

Let X be a smooth complete toric variety. We describe the Altmann-Ilten-Vollmert equivariant deformations of toric varieties in the language of Cox rings. More precisely we construct one parameters…

Log terminal singularities, platonic tuples and iteration of Cox rings

- Mathematics
- 2017

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…

On homogeneous locally nilpotent derivations of trinomial algebras

- MathematicsJournal of Algebra and Its Applications
- 2019

We provide an explicit description of homogeneous locally nilpotent derivations of the algebra of regular functions on affine trinomial hypersurfaces. As an application, we describe the set of roots…

The automorphism group of a rigid affine variety

- Mathematics
- 2016

An irreducible algebraic variety X is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group Aut(X) of a rigid affine variety contains…

Properties of Multihomogeneous Spaces and relation with T-varieties

- Mathematics
- 2021

We study multihomogeneous spaces corresponding to Zn-graded algebras over an algebraically closed field of characteristic 0 and their relation with the description of T -varieties.

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