A geometric characterization of toric varieties

Brown, Morgan V.; McKernan, James; Svaldi, Roberto; Zong, Hong R.

doi:10.1215/00127094-2017-0047

Cited by 19 publications

(16 citation statements)

References 27 publications

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“…This suffices to prove item from (1) to (4). For (5), let W be the canonical ambient toric variety of X . By [15, Prop.…”

Section: Birational Geometry Of Complete Weak Mori Dream Spacesmentioning

confidence: 99%

“…[1], Lemma 1.4.3.1). With respect to the Cl(X)-graded decompositions given in(5), the exact sequence (4) behaves coherently, that is I χ is a Cl(X)-homogeneous sheaf of ideals in S admitting a well-defined Cl(X)-graded decomposition I χ = δ∈Cl(X) I χ,δ such that Cox δ ∼ = S δ /I χ,δ . For every D, D ∈ K, D ∼ D if and only if there exists an isomorphism ψ : H 0 (X, O X (D))…”

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confidence: 99%

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Embedding non-projective Mori dream space

Rossi

2019

Geom Dedicata

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This paper is devoted to extend some Hu-Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, Q-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu-Keel. On the other hand, an example of a wMDS does not admitting any neat embedded sharp completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS.Moreover, termination of the Mori minimal model program (MMP) for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS.Finally, we give a characterization of a wMDS arising from a small Qfactorial modification of a projective weak Q-Fano variety.

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“…This suffices to prove item from (1) to (4). For (5), let W be the canonical ambient toric variety of X . By [15, Prop.…”

Section: Birational Geometry Of Complete Weak Mori Dream Spacesmentioning

confidence: 99%

mentioning

confidence: 99%

Embedding non-projective Mori dream space

Rossi

2019

Geom Dedicata

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“…We explain by an example that what kind of T we will consider. Let us consider weighted hypersurfaces of degree 9 in P = P (1,1,1,3,4). Let x, y, z, w, t be the homogeneous coordinates of weight 1, 1, 1, 3, 4, respectively.…”

Section: Definition 21mentioning

confidence: 99%

Stable rationality of orbifold Fano 3-fold hypersurfaces

Okada

2018

J. Algebraic Geom.

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We determine the rationality of very general quasismooth Fano 3fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.

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“…One direct result of our main theorem is as follows. We refer to [3] for the toric pair. Remark 1.2.…”

Section: Introductionmentioning

confidence: 99%

Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties

Zhong

2020

J Geom Anal

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We consider an arbitrary int-amplified surjective endomorphism f of a normal projective variety X over C and its f −1 -stable prime divisors. We extend the early result in [27, Theorem 1.3] for the case of polarized endomorphisms to the case of intamplified endomorphisms.Assume further that X has at worst Kawamata log terminal singularities. We prove that the total number of f −1 -stable prime divisors has an optimal upper bound dim X + ρ(X), where ρ(X) is the Picard number. Also, we give a sufficient condition for X to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point.2010 Mathematics Subject Classification. 14E30, 32H50, 08A35, Key words and phrases. amplified endomorphism, minimal model program, rationally connected variety.for any m > 0. The first equality is due to projection formula. By definition, the Iitaka dimension κ(X, F 1 ) = 0, a contradiction. Therefore, Claim 3.5 holds.By Claim 3.5, Equation (11) and (12), E 1 = E 2 = ∆ f = 0 and K X + j V j ∼ Q 0.Back to Equation (7), since ∆ f = 0, the ramification divisor consists of only these V j 's.By the purity of branch loci, f isétale outside ( V j ) ∪ f −1 (SingX), which completes the proof of Theorem 1.1 (4) for the case when dim Y 0 = 0. 142, arXiv:0408301.

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A geometric characterization of toric varieties

Abstract: We prove a conjecture of Shokurov which characterises toric varieties using log pairs.

Cited by 19 publications

References 27 publications

Embedding non-projective Mori dream space

Embedding non-projective Mori dream space

Stable rationality of orbifold Fano 3-fold hypersurfaces

Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties

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