Derived categories of nodal del Pezzo threefolds

Pavic, Nebojsa; Shinder, Evgeny

doi:10.48550/arxiv.2108.04499

Cited by 2 publications

(3 citation statements)

References 18 publications

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“…Therefore, we concentrate on the case of quintic del Pezzo threefolds. Similar results in terms of Kawamata decompositions have been obtained by [Xie23] and [PS21a].…”

Section: Threefoldssupporting

confidence: 88%

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Categorical absorptions of singularities and degenerations

Kuznetsov,

Shinder

2024

Épijournal De Géométrie Algébrique

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We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$.

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“…Therefore, we concentrate on the case of quintic del Pezzo threefolds. Similar results in terms of Kawamata decompositions have been obtained by [Xie23] and [PS21a].…”

Section: Threefoldssupporting

confidence: 88%

“…Conversely, a Kawamata semiorthogonal decomposition provides absorption of singularities. For examples of Kawamata decompositions, see [KPS21,PS21a,Xie23].…”

Section: Absorption and Deformation Absorptionmentioning

confidence: 99%

Categorical absorptions of singularities and degenerations

Kuznetsov,

Shinder

2024

Épijournal De Géométrie Algébrique

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“…The applications in §7.3 are related to the study of derived categories of nodal curves in [Bur,KPS21] and nodal threefolds in [Ku21,Ka19,X21,PS21].…”

Section: −−−−→ O ⊕3mentioning

confidence: 99%

Derived projectivizations of complexes

Jiang¹

2022

Preprint

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In this paper, we study the counterpart of Grothendieck's projectivization construction in the context of derived algebraic geometry. Our main results are as follows: First, we define the derived projectivization of a connective complex, study its fundamental properties such as finiteness properties and functorial behaviors, and provide explicit descriptions of their relative cotangent complexes. We then focus on the derived projectivizations of complexes of perfect-amplitude contained in [0, 1]. In this case, we prove a generalized Serre's theorem, a derived version of Beilinson's relations, and establish semiorthogonal decompositions for their derived categories. Finally, we show that many moduli problems fit into the framework of derived projectivizations, such as moduli spaces that arise in Hecke correspondences. We apply our results to these situations. 7.1. The Fourier-Mukai functors and semiorthogonal decompositions 69 7.2. The local situation 72 7.3. Applications to classical examples 75 8. Hecke correspondence moduli as derived projectivizations 84 8.1. Hecke correspondences: the case of one-step flags 85 8.2. Hecke correspondences: the case of general flags 88 8.3. Hecke correspondences for surfaces 91 Appendix A. Simplicial Models for Symmetric and Koszul Algebras 95 A.1. Dold-Puppe and Quillen's construction 95 A.2. Simplicial symmetric algebras 95 A.3. Simplicial Koszul algebras 96 References 98

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Derived categories of nodal del Pezzo threefolds

Cited by 2 publications

References 18 publications

Categorical absorptions of singularities and degenerations

Categorical absorptions of singularities and degenerations

Derived projectivizations of complexes

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