Rouquier dimension is Krull dimension for normal toric varieties

Favero, David; Huang, Jesse

doi:10.48550/arxiv.2302.09158

Cited by 2 publications

(2 citation statements)

References 11 publications

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“…Remark 1.7. An independent and concurrent proof of Corollary E using homological mirror symmetry and ideas similar to those explained in Section 1.2 was obtained by Favero and Huang in [FH23]. We would also like to note that Ballard previously suggested that toric Frobenius should be useful for resolving Orlov's conjecture for toric varieties.…”

Section: Resultsmentioning

confidence: 68%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

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Given any toric subvariety Y of a smooth toric variety X of codimension k, we construct a length k resolution of O Y by line bundles on X. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of O X under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry.As a corollary, we obtain a virtual analogue of Hilbert's syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman, and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal's claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara's conjecture that this remains true for arbitrary line bundles.

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Section: Resultsmentioning

confidence: 68%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

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“…In passing we note that the same method also applies to the toric case provided the anti-canonical line bundle −K X is nef. In fact, Orlov's conjecture holds for any toric variety, see the recent preprint [FH23].…”

Section: Introductionmentioning

confidence: 99%

Exceptional sequences of line bundles on projective bundles

Altmann¹,

Hochenegger²,

Witt³

2023

Preprint

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For a vector bundle E → P ℓ we investigate exceptional sequences of line bundles on the total space of the projectivisation X = P(E). In particular, we consider the case of the cotangent bundle of P ℓ . If ℓ = 2, we will completely classify the (strong) exceptional sequences and show that any maximal exceptional sequence is full. For general ℓ, we prove that the Rouquier dimension of D(X) equals dim X, thereby confirming a conjecture of Orlov.

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Rouquier dimension is Krull dimension for normal toric varieties

Cited by 2 publications

References 11 publications

Resolutions of toric subvarieties by line bundles and applications

Resolutions of toric subvarieties by line bundles and applications

Exceptional sequences of line bundles on projective bundles

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