A counterexample to King's conjecture

Hille, Lutz; Perling, Markus

doi:10.1112/s0010437x06002260

Cited by 53 publications

(43 citation statements)

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“…Kawamata [2006] showed that every smooth, projective toric variety admits a full exceptional collection, but not necessarily a strong one. Hille and Perling [2006] have recently constructed a toric counterexample to King's conjecture, but the question of how common such collections are is still wide open. Example 1.12.…”

Section: Bondal Quiversmentioning

confidence: 99%

Moduli spaces for Bondal quivers

Bergman¹,

Proudfoot²

2008

Pacific J. Math.

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Given a sufficiently nice collection of sheaves on an algebraic variety V , Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V . We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory.

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Section: Bondal Quiversmentioning

confidence: 99%

Moduli spaces for Bondal quivers

Bergman¹,

Proudfoot²

2008

Pacific J. Math.

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“…Kawamata [Kaw06] showed that every smooth toric Deligne-Mumford stack has a full exceptional collection of sheaves, but we note that these collections are not shown to be strong, nor do they consist of bundles. It is important to note that the existence of full strong exceptional collections of line bundles is rare; Hille-Perling [HP06] constructed smooth toric surfaces that do not have such collections. Even when only considering smooth toric Fano varieties, there exist examples in dimensions ≥ 419 that do not have full strong exceptional collections of line bundles, as demonstrated by Efimov [Efi10].…”

Section: Introductionmentioning

confidence: 99%

Tilting bundles on toric Fano fourfolds

Prabhu-Naik

2017

Journal of Algebra

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Abstract. This paper constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth 4-dimensional toric Fano varieties. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolledup helix algebra. A database of the full strong exceptional collections can be found in the package QuiversToricVarieties for the computer algebra system Macaulay2.

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“…There is an example of a smooth toric surface which does not admit a full strong exceptional collection of line bundles, see [9]. A quick review of the related results can be found in [7].…”

Section: Remark 310mentioning

confidence: 99%

“…While this turned out to be false, see [9], it is still natural to conjecture that every smooth nef-Fano toric variety possesses such a collection, and there is some numerical evidence towards it. Here a variety is called nef-Fano (also often referred to as weak Fano) if its anticanonical class is nef and big, though not necessarily ample.…”

Section: Introductionmentioning

confidence: 94%