On birational rigidity of singular del Pezzo fibrations of degree 1

Okada, Takuzo

doi:10.1112/jlms.12304

Cited by 4 publications

(3 citation statements)

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“…The -condition (1.1) implies that is not in the movable cone of and, thus, . By Proposition 6.2 and [Oka20a, Proposition 2.7], there are points in distinct -fibers and positive rational numbers such that are centers of non-canonical singularities of the pair and , where is the -fiber containing .…”

Section: Del Pezzo Fibrations Of Degreementioning

confidence: 99%

“…By Corti's inequality Theorem 3.17, there are numbers , , with such that We have and, thus, Note that is isomorphic to an irreducible and reduced weighted hypersurface of degree in . By [Oka20a, Lemma 2.10], we can take a curve which passes through and which does not contain any component of . Then we can take a divisor , where is a sufficiently large integer, such that .…”

Section: Del Pezzo Fibrations Of Degreementioning

confidence: 99%

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2 n²-inequality for cA₁ points and applications to birational rigidity

Krylov,

Okada,

Paemurru

et al. 2024

Compositio Math.

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The $4 n^2$ -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$ , and obtain a $2 n^2$ -inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$ -condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.

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Section: Del Pezzo Fibrations Of Degreementioning

confidence: 99%

Section: Del Pezzo Fibrations Of Degreementioning

confidence: 99%

2 n²-inequality for cA₁ points and applications to birational rigidity

Krylov,

Okada,

Paemurru

et al. 2024

Compositio Math.

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show abstract

“…The aim of this section is to prove the following by the method of maximal singularities developed in [Puk98] and [Oka20a] combined with Corti's inequality for smooth points and cA 1 points. Theorem 6.1.…”

Section: Del Pezzo Fibrations Of Degreementioning

confidence: 99%

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity

Krylov¹,

Okada²,

Paemurru³

et al. 2022

Preprint

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The 4n 2 -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type cA1, and obtain a 2n 2 -inequality for cA1 points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree 1 over P 1 satisfying the K 2 -condition, all of which have at most terminal cA1 singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a cA1 point which is not an ordinary double point.

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Birational Rigidity of Orbifold Degree 2 Del Pezzo Fibrations

Ahmadinezhad¹,

Krylov²

2022

Nagoya Math. J.

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Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$ -condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for most families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalized $K^2$ -condition are birationally rigid.

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On birational rigidity of singular del Pezzo fibrations of degree 1

Cited by 4 publications

References 20 publications

2 n²-inequality for cA₁ points and applications to birational rigidity

2 n²-inequality for cA₁ points and applications to birational rigidity

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity

Birational Rigidity of Orbifold Degree 2 Del Pezzo Fibrations

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On birational rigidity of singular del Pezzo fibrations of degree 1

Cited by 4 publications

References 20 publications

2 n2-inequality for cA1 points and applications to birational rigidity

2 n2-inequality for cA1 points and applications to birational rigidity

$2 n^2$-inequality for $cA_1$ points and applications to birational rigidity

Birational Rigidity of Orbifold Degree 2 Del Pezzo Fibrations

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2 n²-inequality for cA₁ points and applications to birational rigidity

2 n²-inequality for cA₁ points and applications to birational rigidity