Normal singularities with torus actions

Liendo, Alvaro; Süß, Hendrik

doi:10.2748/tmj/1365452628

Cited by 38 publications

(57 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…[9] and Gongyo et al [17] and factorial by [7]. Hence, by Remark 6.4 in [25] we obtain the last statement. Since L is a sum of at least three monomials A i , whose partial derivatives also vanish, the statement follows.…”

Section: Pv and μ(V) Denotes The Minimal Positive Integer Such That supporting

confidence: 65%

Flexible affine cones and flexible coverings

2018

View full text Add to dashboard Cite

show abstract

Section: Pv and μ(V) Denotes The Minimal Positive Integer Such That supporting

confidence: 65%

Flexible affine cones and flexible coverings

2018

View full text Add to dashboard Cite

show abstract

“…Note that this is always the case if X(S) is smooth; indeed this follows from [LS10], proposition 5.1 and theorem 5.3 together with the fact that height-one hyperplane sections of smooth cones only have trivial admissible Minkowski decompositions. To any locally trivial oneparameter deformation of X(S) we can assign a class in H 1 (X(S), T X(S) ) via the Kodaira-Spencer map: we pull back the deformation to one over Spec C[t]/t 2 , and the Kodaira-Spender correspondence gives a bijection between such first-order deformations and the above-mentioned cohomology classes.…”

Section: Locally Trivial Deformationsmentioning

confidence: 90%

“…This follows from the description of the general fiber from theorem 2.8 coupled with [LS10], theorem 5.3. Proof.…”

Section: Deformations Of Rational T -Varieties 541mentioning

confidence: 99%

See 1 more Smart Citation

Deformations of rational 𝑇-varieties

Ilten

Vollmert²

2011

J. Algebraic Geom.

View full text Add to dashboard Cite

We show how to construct certain homogeneous deformations for rational normal varieties with codimension one torus action. This can then be used to construct homogeneous deformations of any toric variety in arbitrary degree. For locally trivial deformations coming from this construction, we calculate the image of the Kodaira-Spencer map. We then show that for a smooth complete toric variety, our homogeneous deformations span the space of first-order deformations.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf NATHAN OWEN ILTEN AND ROBERT VOLLMERTThe goal of this paper is to generalize the results of [Ilt10b] in several directions. First of all, we construct multi-parameter deformations of varieties of arbitrary dimension which are also not necessarily smooth. Secondly, we will look not only at deformations of toric varieties but also deformations of rational T -varieties of complexity one, that is, rational normal varieties admitting an effective codimension one torus action.Much as an n-dimensional toric variety can be described by an ndimensional fan, an n-dimensional T -variety X of complexity one can be described by a curve and some n − 1-dimensional combinatorial data. We then construct a deformation of X by somehow deforming the corresponding combinatorial data. In section 1, we give a short overview of the necessary theory of T -varieties. We then show how to construct homogeneous deformations of affine T -varieties in section 2. Here we also describe the fibers of such deformations explicitly as T -varieties. Note that the deformation theory of affine T -varieties is being further developed by the second author in [Vol10]. As a special case, we can of course consider toric varieties with an action by some subtorus. We describe this in detail in section 3 and show how to recover the deformations constructed by Altmann. In particular, we have a very natural description of toric deformations with non-negative degree, which are essential for constructing homogeneous deformations of complete toric varieties.In section 4 we then show how to glue the deformations of affine T -varieties together to construct deformations of non-affine T -varieties. As in the affine case, we can also describe the fibers of such deformations explicitly as Tvarieties. Restricting to the case of locally trivial deformations, we then calculate the Kodaira-Spencer map in section 5.Of course, non-affine toric varieties again provide an example where our construction can be put to use. In section 6 we reformulate our Kodaira-Spencer calculation in nicer terms for this special case. For a smooth complete toric variety X, we then construct certain special deformations and show that they in fact span T 1 X . Thus, at least for smooth complete toric varieties, our deformations provide a kind of skeleton of the versal deformation.Our approach has some aspects in common with the independent work of Mavlyutov; both approaches construct deformations, via Minkowski decomposition,...

show abstract

“…In Section 3, we will use this procedure to compute the local models for G m -threefolds. Here, we now give as a simple corollary the well known criterion for the smoothness of a complexity one T-variety [LS13]. (ii) Y = P 1 and X is the affine space endowed with a linear T-action.…”

Section: Smoothness Criteriamentioning

confidence: 99%

Smooth varieties with torus actions

Liendo

Petitjean

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the étale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the étale topology to the combinatorial data describing affine space endowed with a linear torus action. Finally, we provide an effective method to check the smoothness of a Gm-threefold in terms of the combinatorial data.

show abstract

Normal singularities with torus actions

Cited by 38 publications

References 26 publications

Flexible affine cones and flexible coverings

Flexible affine cones and flexible coverings

Deformations of rational 𝑇-varieties

Smooth varieties with torus actions

Contact Info

Product

Resources

About