Semiorthogonal decompositions in families

Кузнецов, А. А.

doi:10.48550/arxiv.2111.00527

Cited by 2 publications

(3 citation statements)

References 41 publications

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“…is the ring of derived dual numbers over κ; These semiorthogonal decompositions agree with the results in [Ku21,Example 5.2,Theorem 5.12].…”

Section: Applications To Classical Examplessupporting

confidence: 85%

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Derived projectivizations of complexes

Jiang¹

2022

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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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“…is the ring of derived dual numbers over κ; These semiorthogonal decompositions agree with the results in [Ku21,Example 5.2,Theorem 5.12].…”

Section: Applications To Classical Examplessupporting

confidence: 85%

“…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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“…There are alternative approaches to some of our results: Theorem 1.4 was proved independently by Zhang in [Zha21], based on a study of Bridgeland moduli spaces, while Theorem 1.6 was proved independently by Kuznetsov and Shinder in work in preparation [KS22] (see also [Kuz21,§5.4]), based on a degeneration argument and a theory of "absorption of singularities". Our paper and these two use completely different methods, which we believe are interesting in their own right.…”

Section: Related Workmentioning

confidence: 99%

Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

Bayer¹,

Alexander²

2022

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We settle the last open case of Kuznetsov's conjecture on the derived categories of Fano threefolds. Contrary to the original conjecture, we prove the Kuznetsov components of quartic double solids and Gushel-Mukai threefolds are never equivalent, as recently shown independently by Zhang. On the other hand, we prove the modified conjecture asserting their deformation equivalence. Our proof of nonequivalence combines a categorical Enriques-K3 correspondence with the Hodge theory of categories, and gives as a byproduct a simple proof of a theorem of Debarre and Kuznetsov about the fibers of the period map for Gushel-Mukai varieties. Our proof of deformation equivalence relies on results of independent interest about obstructions to enhancing group actions on categories.

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Semiorthogonal decompositions in families

Cited by 2 publications

References 41 publications

Derived projectivizations of complexes

Derived projectivizations of complexes

Kuznetsov's Fano threefold conjecture via K3 categories and enhanced group actions

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