Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Auel, Asher; Bernardara, Marcello; Bolognesi, Michele

doi:10.1016/j.matpur.2013.11.009

Cited by 56 publications

(122 citation statements)

References 70 publications

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“…The theory of exceptional objects and semiorthogonal decompositions in the case where

k

is algebraically closed and of characteristic 0 was studied in the Rudakov seminar at the end of the 80s, and developed by Rudakov, Gorodentsev, Bondal, Kapranov, and Orlov among others, see . As noted in , most fundamental properties persist over any base field

k

.…”

Section: Semiorthogonal Decompositions and Categorical Representabilitymentioning

confidence: 99%

“…Based on base‐change results for Fourier–Mukai functor due to Orlov (see ), the following statement was proved in [, Lemma 2.9] (see also [, Proposition 2.1]). Lemma Let

X

be a smooth projective variety over

k

and let

K

be a finite field extension of

k

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

“…Other known (similar) examples include the decomposition of a relative Severi–Brauer variety whose base‐change is the decomposition of a projective bundle given by Orlov , the decomposition of a generalized Severi–Brauer variety given by Blunk whose base‐change is the decomposition of a Grassmannian variety given by Kapranov , and the decomposition of a quadric hypersurface given in (generalizing the work of Kuznetsov ) whose base‐change is the decomposition of a quadric given by Kapranov . Remark One might wonder, motivated by the previous examples, whether the descent of an exceptional block

E \subset {sans-serifD}^{normalb} false(X_{k^{s}} false)

, generated by indecomposable vector bundles

V_{i}

can be realized by the descent of the tilting bundle

V = ⨁_{i} V_{i}

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

“…This question, which was the original motivation for the current work, is now central to a growing body of research into arithmetic aspects of the theory of derived categories, see . As an example, if

S

is a smooth projective surface, Hassett and Tschinkel [, Lemma 8] prove that the index of

S

can be recovered from

{sans-serifD}^{normalb} false(S false)

as the greatest common divisor of the second Chern classes of objects.…”

Section: Introductionmentioning

confidence: 99%

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Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Auel

Bernardara

2018

Proc. London Math. Soc.

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We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary field, namely, that its derived category decomposes into zero‐dimensional components. When S has degree at least 5 we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

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“…The theory of exceptional objects and semiorthogonal decompositions in the case where

k

k

.…”

Section: Semiorthogonal Decompositions and Categorical Representabilitymentioning

confidence: 99%

“…Based on base‐change results for Fourier–Mukai functor due to Orlov (see ), the following statement was proved in [, Lemma 2.9] (see also [, Proposition 2.1]). Lemma Let

X

be a smooth projective variety over

k

and let

K

be a finite field extension of

k

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

E \subset {sans-serifD}^{normalb} false(X_{k^{s}} false)

, generated by indecomposable vector bundles

V_{i}

can be realized by the descent of the tilting bundle

V = ⨁_{i} V_{i}

.…”

Section: Descent For Semiorthogonal Decompositionsmentioning

confidence: 99%

S

is a smooth projective surface, Hassett and Tschinkel [, Lemma 8] prove that the index of

S

can be recovered from

{sans-serifD}^{normalb} false(S false)

as the greatest common divisor of the second Chern classes of objects.…”

Section: Introductionmentioning

confidence: 99%

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Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields

Auel

Bernardara

2018

Proc. London Math. Soc.

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“…Smoothness of D and X implies that π has simple degeneration; see [HVV11, Rem. 7.1] or [ABB13,Prop. 1.6].…”

mentioning

confidence: 99%

Cubic fourfolds containing a plane and a quintic del Pezzo surface

et al. 2014

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We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C 8 ∩ C 14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.

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