Singularities of Cox rings of Fano varieties

“…Z/2Z × Z/2Z 0 1 0 1 0 1 1 0 8.1 (6) Z/3Z 1 2 0 0 8.1 (7) Z/2Z 1 0 1 0 8.1 (8) {0} -8.3 (10-e) Z 0 1 −2 −1 2 8.1 (10-o) {0} -8.1 (11) Z/2Z 0 1 0 1 8.1 (12- (14) Z/3Z 1 2 0 0 8.1 (15,17,18) {0} -8.1 (16) Z/2Z 1 0 0 1…”

Section: (5-o)mentioning

confidence: 99%

Log terminal singularities, platonic tuples and iteration of Cox rings

Arzhantsev

Braun

Hausen

et al. 2017

European Journal of Mathematics

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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 group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions.As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.Theorem 2. Let X 1 be a rational, normal, affine variety with a torus action of complexity one of Type 2 and at most log terminal singularities. Then there is a unique chain of quotients

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“…Then X is of Fano type (resp., Calabi-Yau type) if and only if Spec(Cox(X)) has klt singularities (resp. log canonical singularities) [KO15] (see also [GOST15], [Bro13]). Recall that X is said to be of Calabi-Yau type if there exists a log-canonical pair (X, ∆) such that (K X + ∆) is Q-linearly trivial.…”

Section: 5mentioning

confidence: 99%

Mori dream spaces and blow-ups

Castravet¹

2018

Algebraic Geometry: Salt Lake City 2015

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The goal of the present article is to survey the general theory of Mori Dream Spaces, with special regards to the question: When is the blow-up of toric variety at a general point a Mori Dream Space? We translate the question for toric surfaces of Picard number one into an interpolation problem involving points in the projective plane. An instance of such an interpolation problem is the Gonzalez-Karu theorem that gives new examples of weighted projective planes whose blow-up at a general point is not a Mori Dream Space.

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Singularities of Cox rings of Fano varieties

Abstract: Let X be a smooth complete Fano variety over C. We show that the Cox ring L∈Pic(X) H 0 (X, OX (L)) is Gorenstein with canonical singularities.

Cited by 19 publications

References 18 publications

Flexible affine cones and flexible coverings

Flexible affine cones and flexible coverings

Log terminal singularities, platonic tuples and iteration of Cox rings

Mori dream spaces and blow-ups

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