Categorical representability and intermediate Jacobians of Fano threefolds

Bernardara, Marcello; Bolognesi, Michele

doi:10.4171/115-1/1

Cited by 19 publications

(31 citation statements)

References 55 publications

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“…Using semiorthogonal decompositions, one can define a notion of categorical representability for a dg‐enhanced triangulated category. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see . Definition A dg‐enhanced

k

‐linear triangulated category

sans-serifT

is representable in dimension

m

if it admits a semiorthogonal decomposition

T = ⟨ {sans-serifA}_{1}, \dots, {sans-serifA}_{r} ⟩,

and for each

i = 1, \dots, r

there exists a smooth projective connected

k

‐variety

Y_{i}

with

\dim Y_{i} ⩽ m

, such that

A_{i}

is equivalent to an admissible subcategory of

{sans-serifD}^{normalb} false(Y_{i} false)

.…”

Section: Semiorthogonal Decompositions and Categorical Representabilitymentioning

confidence: 99%

“…Using semiorthogonal decompositions, one can define a notion of categorical representability for a dg-enhanced triangulated category. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [22]. Definition 1.14 [22].…”

Section: Categorical Representabilitymentioning

confidence: 99%

“…In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [22]. Definition 1.14 [22]. A dg-enhanced k-linear triangulated category T is representable in dimension m if it admits a semiorthogonal decomposition…”

Section: Categorical Representabilitymentioning

confidence: 99%

“…Definition 1.15 [22]. Let X be a projective k-variety of dimension n. We say that X is categorically representable in dimension m (or equivalently in codimension n − m) if there exists a categorical resolution of singularities of D b (X) that is representable in dimension m.…”

Section: Categorical Representabilitymentioning

confidence: 99%

“…In this context, based on the classical notion of representability for Chow groups and motives, Bolognesi and the second author proposed the notion of categorical representability (see Definition ), and formulated the following question. Question Is a rational variety always categorically representable in codimension 2?…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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