Birational rigidity of orbifold degree 2 del Pezzo fibrations

Abban, Hamid; Krylov, Igor

doi:10.48550/arxiv.1710.05328

Cited by 3 publications

(12 citation statements)

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“…Suppose that X/P 1 does not satisfy the K 2 -condition. Then (λ, µ, ν) ∈ {(0, −2, 0), (0, −1, 0), (0, −1, 1), (0, 0, 1), (1,1,3), (1,2,4), (2,3,6)}.…”

Section: Del Pezzo Fibrations Embedded As a Hypersurface In A Toricmentioning

confidence: 99%

“…A del Pezzo fibration X → P 1 over P 1 is said to satisfy the K 2 -condition (resp. K-condition) if the 1-cycle (−K X ) 2 is not contained in the interior of the cone NE(X) of effective curves on X (resp. −K X is not in the interior of the movable cone Mov(X)).…”

Section: Introductionmentioning

confidence: 99%

“…In the context of Minimal Model Program, it is natural and important to study singular del Pezzo fibrations. Recently there are some progress for singular del Pezzo fibrations of degree 2: Krylov [15] and Ahmadinezhad-Krylov [2] proved that a del Pezzo fibration of degree 2 with only singular points of type 1 2 (1, 1, 1) satisfying the K 2 -condition is birationally rigid under some additional assumptions.…”

Section: Introductionmentioning

confidence: 99%

“…We see that the K 2 -condition is equivalent to the K 3 0 -condition. Indeed, since the nef cone of X is spanned by F and −K X + nef(X/P 1 )F , and F • (−K X ) 2 = 1 > 0, the 1-cycle (−K X ) 2 is not in the interior of NE(X), i.e., the K 2 -condition is satisfied, if and only if the inequality (−K X + nef(X/P 1 )F ) • (−K X ) 2 ≤ 0 holds, which is nothing but the K 3 0 -condition. Moreover it is obvious that the K 3 δcondition implies the K 3 δ -condition for δ ≤ δ .…”

Section: Introductionmentioning

confidence: 99%

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On birational rigidity of singular del Pezzo fibrations of degree 1

Okada

2020

J. London Math. Soc.

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We give a sufficient condition for birational superrigidity of del Pezzo fibrations of degree 1 with only 1 2 (1, 1, 1) singular points, generalizing the so called K 2 -condition. As an application, we also prove that a del Pezzo fibrations of degree 1 with only 1 2 (1, 1, 1) singular points embedded in a toric P(1, 1, 2, 3)-bundle over P 1 is birationally superrigid if and only if it satisfies the K-condition.Definition 1.2. Let X/P 1 be a del Pezzo fibration of degree 1. We denote by F the fiber class of the fibration X → P 1 . We define nef(X/P 1 ) := inf{ r | −K X + rF is nef }, 2010 Mathematics Subject Classification. 14E07 and 14E08 and 14J30.

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“…Suppose that X/P 1 does not satisfy the K 2 -condition. Then (λ, µ, ν) ∈ {(0, −2, 0), (0, −1, 0), (0, −1, 1), (0, 0, 1), (1,1,3), (1,2,4), (2,3,6)}.…”

Section: Del Pezzo Fibrations Embedded As a Hypersurface In A Toricmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On birational rigidity of singular del Pezzo fibrations of degree 1

Okada

2020

J. London Math. Soc.

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“…The quadratic technique goes back to the paper of V.A.Iskovskikh and Yu.I.Manin on the three-dimensional quartic [21]; almost all results on the birational rigidity in the absolute case and for Fano-Mori fibre spaces over P 1 were obtained by means of that technique. Among the recent papers, where it is applied in the proof of birational rigidity, we mention [22,23,24]. In [25,5] both techniques were used.…”

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confidence: 99%

Birational geometry of varieties, fibred into complete intersections of codimension two

Pukhlikov¹

2021

Preprint

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In this paper we prove the birational superrigidity of Fano-Mori fibre spaces π: V → S, every fibre of which is a complete intersection of type d 1 • d 2 in the projective space P d 1 +d 2 , satisfying certain conditions of general position, under the assumption that the fibration V /S is sufficiently twisted over the base (in particular, under the assumption that the K-condition holds). The condition of general position for every fibre guarantees that the global log canonical threshold is equal to one. This condition bounds the dimension of the base S by a constant that depends on the dimension M of the fibre only (as the dimension M of the fibre grows, this constant grows as 1 2 M 2 ). The fibres and the variety V itself may have quadratic and bi-quadratic singularities, the rank of which is bounded from below.Bibliography: 34 items.

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Birationally rigid Fano fibre spaces. II

Pukhlikov¹

2015

Izv. Math.

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In this paper we prove birational rigidity of large classes of Fano-Mori fibre spaces over a base of arbitrary dimension, bounded from above by a constant that depends on the dimension of the fibre only. In order to do that, we first show that if every fibre of a Fano-Mori fibre space satisfies certain natural conditions, then every birational map onto another Fano-Mori fibre space is fibre-wise. After that we construct large classes of fibre spaces (into Fano double spaces of index one and into Fano hypersurfaces of index one) which satisfy those conditions. Bibliography: 35 titles.

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Birational rigidity of orbifold degree 2 del Pezzo fibrations

Cited by 3 publications

References 0 publications

On birational rigidity of singular del Pezzo fibrations of degree 1

On birational rigidity of singular del Pezzo fibrations of degree 1

Birational geometry of varieties, fibred into complete intersections of codimension two

Birationally rigid Fano fibre spaces. II

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