Cubic fourfolds containing a plane and a quintic del Pezzo surface

Auel, Asher; Bernardara, Marcello; Bolognesi, Michele; Várilly-Alvarado, Anthony

doi:10.14231/ag-2014-010

Cited by 32 publications

(46 citation statements)

References 24 publications

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“…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%

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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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%

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

Auel

Bernardara

2018

Proc. London Math. Soc.

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“…Some recent results involving cubic fourfolds containing a plane that make use of the tool of quadric fibrations can be found in [3], [5], [11], [22], [23], [25] and [27].…”

Section: 2mentioning

confidence: 99%

The derived category of a non-generic cubic fourfold containing a plane

Moschetti

2018

Mathematical Research Letters

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We describe an Azumaya algebra on the resolution of singularities of the double cover of a plane ramified along a nodal sextic associated to a non generic cubic fourfold containing a plane. We show that the derived category of such a resolution, twisted by the Azumaya algebra, is equivalent to the Kuznetsov component in the semiorthogonal decomposition of the derived category of the cubic fourfold.

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“…A number of recent papers have explored smooth limits of Pfaffian cubic fourfolds more systematically. For analysis of the intersection between cubic fourfolds containing a plane and limits of the Pfaffian locus, see [ABBVA14]. Auel and Bolognese-Russo-Staglianò [BRS15] have shown that smooth limits of Pfaffian cubic fourfolds are always rational; [BRS15] includes a careful analysis of the topology of the Pfaffian locus in moduli.…”

Section: Cubic Fourfolds Containing a Quintic Del Pezzo Surfacementioning

confidence: 99%

Cubic Fourfolds, K3 Surfaces, and Rationality Questions

Hassett

2016

Lecture Notes in Mathematics

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This is a survey of the geometry of complex cubic fourfolds with a view toward rationality questions. Smooth cubic surfaces have been known to be rational since the 19th century [Dol05]; cubic threefolds are irrational by the work of Clemens and Griffiths [CG72]. Cubic fourfolds are likely more varied in their behavior. While there are examples known to be rational, we expect that most cubic fourfolds should be irrational. However, no cubic fourfolds are proven to be irrational.Our organizing principle is that progress is likely to be driven by the dialectic between concrete geometric constructions (of rational, stably rational, and unirational parametrizations) and conceptual tools differentiating various classes of cubic fourfolds (Hodge theory, moduli spaces and derived categories, and decompositions of the diagonal). Thus the first section of this paper is devoted to classical examples of rational parametrizations. In section two we focus on Hodge-theoretic classifications of cubic fourfolds with various special geometric structures. These are explained in section three using techniques from moduli theory informed by deep results on K3 surfaces and their derived categories. We return to constructions in the fourth section, focusing on unirational parametrizations of special classes of cubic fourfolds. In the last section, we touch on recent applications of decompositions of the diagonal to rationality questions, and what they mean for cubic fourfolds.

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Cubic fourfolds containing a plane and a quintic del Pezzo surface

Cited by 32 publications

References 24 publications

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

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

The derived category of a non-generic cubic fourfold containing a plane

Cubic Fourfolds, K3 Surfaces, and Rationality Questions

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