The toric Frobenius morphism and a conjecture of Orlov

Ballard, Matthew; Duncan, Alexander; McFaddin, Patrick K.

doi:10.1007/s40879-018-0266-5

Cited by 5 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the general result remains elusive. Restricting to the toric setting, a notable result is [BDM19] which verifies the conjecture when a certain subset of Θ gives rise to a tilting object and for all smooth toric Fano threefolds and fourfolds. When X is proper, the length of a resolution of the diagonal gives an upper bound on the generation time of the corresponding classical generator.…”

Section: Resultssupporting

confidence: 60%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Resultssupporting

confidence: 60%

Resolutions of toric subvarieties by line bundles and applications

Hanlon¹,

Hicks²,

Lazarev³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…As it turns out, the summands of the probe generator correspond to a natural stratification of the torus first considered by Bondal-Ruan [Bon06] who established a bijection between strata and summands of the toric Frobenius pushforward of O. Therefore, unsurprisingly, we find that the probe generator is taken to the toric Frobenius pushforward of O (the same generator studied in [BDM19]).…”

Section: Introductionmentioning

confidence: 56%

Rouquier dimension is Krull dimension for normal toric varieties

Favero¹,

Huang²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Special cases of Theorem 1.5 had been established in [BC23, BF12, BDM19, BFK19] before Favero-Huang and Hanlon-Hicks-Lazarev proved it in general. The full version of Orlov’s Conjecture states that Theorem 1.5 extends to any smooth quasi-projective variety; see [BC23, §1.2] for a list of known cases of this conjecture.…”

Section: Resultsmentioning

confidence: 99%