Derived category of projectivizations and flops

Jiang, Qingyuan; Leung, Naichung Conan

doi:10.48550/arxiv.1811.12525

Cited by 14 publications

(48 citation statements)

References 54 publications

(116 reference statements)

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“…The conjecture in the case d = 1 is equivalent to the projectivization formula D(P(G )) = D(P(K )), δ-copies of D(X) , proved by the author and Leung in [JL18]; see [JL18] or Thm. 6.16 for more details.…”

Section: Introductionmentioning

confidence: 95%

“…For example, in the projectivization formula [JL18] mentioned above, the last component "δ-copies of D(X)" could be induced by either of the following relative exceptional sequences:…”

Section: Introductionmentioning

confidence: 99%

“…(3) The projectivization formula [JL18] of the author and Leung; see Thm. 6.16; (4) Pirozhkov's formula [Pi20] for generalized Cayley's trick; see Thm.…”

Section: Introductionmentioning

confidence: 99%

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Derived categories of Quot schemes of locally free quotients, I

Jiang¹

2021

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This paper studies the derived category of the Quot scheme of rank d locally free quotients of a sheaf G of homological dimension ≤ 1 over a scheme X. In particular, we propose a conjecture about the structure of its derived category and verify the conjecture in various cases. This framework allows us to relax certain regularity conditions on various known formulae -such as the ones for blowups (along Koszul-regular centers), Cayley's trick, standard flips, projectivizations, and Grassmannain-flips -and supplement these formulae with the results on mutations and relative Serre functors. This framework also leads us to many new phenomena such as virtual flips, and structural results for the derived categories of (i) Quot 2 schemes, (ii) flips from partial desingularizations of rank ≤ 2 degeneracy loci, and (iii) blowups along determinantal subschemes of codimension ≤ 4. Generalities on derived categoriesof schemes 3.2. Generators of triangulated categories 3.3. Semiorthogonal decompositions and mutations 3.4. Postnikov systems and convolutions 3.5. Closed monoidal structures 3.6. Linear categories 3.7. Relative Serre functors 3.8. Base change of linear categories 3.9. Relative Fourier-Mukai transforms 3.10. Compositions of relative Fourier-Mukai transforms 3.11. Relative exceptional collections 3.12. Grassmannian bundles Part II. Local geometry 4. Young diagrams and Grassmannians 4.1. Young diagrams and Schur functors 4.2. Borel-Bott-Weil theorem and Kapranov's collections 4.3. Mutations on Grassmannians 5. Local geometry and correspondences 5.1. The key lemma and Lascoux-type resolutions 5.2. First implications 5.3. The case d + = 1: projectivization 5.4. The case ℓ + = 1: standard flips 5.5. The case m = 1: Pirozhkov's theorem 5.6. The cases δ ≤ 3 5.7. The case d + = 2: Quot 2 -formula 5.8. The case ℓ + = 2: flips from resolving rank ≤ 2 degeneracy lociPart III. Global geometry 6. Global situation 6.1. Hom spaces 6.2. Tor-independent conditions and general procedures of base-change 6.3. Blowups along Koszul-regularly immersed centers 6.4. Cayley's trick 6.5. Projectivizations 6.6. Generalized Caylay's trick and Pirozhkov's theorem 6.7. Quot 2 -formula 108 7. Flips, flops and virtual flips 110 7.1. First results: Grassmannian flips and virtual flips 110 7.2. Standard flips revisited 112 7.3. Flips from partial desingularizations of rank ≤ 2 degeneracy loci 114 7.4. The cases rank G ≤ 3, and blowups of determinantal ideals of height ≤ 4 115 Appendix A. Relations in the Grothendieck rings of varieties 121 Appendix B. Characteristic-free results for projective bundles 123 References 127

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Section: Introductionmentioning

confidence: 95%

“…For example, in the projectivization formula [JL18] mentioned above, the last component "δ-copies of D(X)" could be induced by either of the following relative exceptional sequences:…”

Section: Introductionmentioning

confidence: 99%

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Derived categories of Quot schemes of locally free quotients, I

Jiang¹

2021

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“…In fact, by [To,Corollary 5.11], there exists a non-trivial semi-orthogonal decomposition of D b (C (n) ), where C is a curve of genus g ≥ 2, if n ≥ g (see also [BK, Theorem D] and [JL,Corollary 3.8]). Note that this is what is expected by Conjecture 1.1, because C (n) is not minimal in the range n ≥ g ≥ 2.…”

Section: 1mentioning

confidence: 99%

Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability of derived categories

Caucci¹

2021

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Motivated by an indecomposability criterion of Xun Lin for the bounded derived category of coherent sheaves on a smooth projective variety X, we study the paracanonical base locus of X, that is the intersection of the base loci of ωX ⊗Pα, for all α ∈ Pic 0 X. We prove that this is equal to the relative base locus of ωX with respect to the Albanese morphism of X. As an application, we get that bounded derived categories of Hilbert schemes of points on certain surfaces do not admit non-trivial semi-orthogonal decompositions. We also have a consequence on the indecomposability of bounded derived categories in families. Finally, our viewpoint allows to unify and extend some results recently appearing in the literature.Conjecture 1.1. Let X be a smooth projective variety. If D b (X) has no non-trivial semiorthogonal decompositions, then X is minimal, i.e., the canonical line bundle ω X is nef.Examples of varieties whose derived categories admit no non-trivial semiorthogonal decompositions are, for instance, varieties with trivial, or, more generally, algebraically trivial canonical bundle ([Br] and [KO, Corollary 1.7], respectively), curves of genus ≥ 1 [Ok], varieties whose Albanese morphism is finite [Pi, Theorem 1.4]. 1 It is well known that the converse direction in Conjecture 1.1 is false (a counterexample is furnished by Enriques surfaces [Zu]), but we have the following folklore (see [BGL, Conjecture 1.6], or [BBOR, Question E]):Conjecture 1.2. Let X be a smooth projective variety. If ω X is nef and effective, then D b (X) has no non-trivial semi-orthogonal decompositions.At the moment of writing, this conjecture (as even the above Conjecture 1.1) is widely open in general and there are just some (classes of) varieties for which it has been verified: see [KO, BBOR], besides the references already quoted above. A particularly interesting case is given by symmetric Key words and phrases. Derived categories, semi-orthogonal decompositions, Albanese morphism, canonical bundle, paracanonical base locus.The author is supported by the ERC Consolidator Grant ERC-2017-CoG-771507-StabCondEn.1 To be precise, in [Pi, Theorem 1.4], Pirozhkov proved that such varieties are noncommutatively stably semiorthogonally indecomposable, which is a stronger notion than indecomposability.

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“…The above result is conjectured by Qingyuan Jiang [Jia,Conjecture A.5]. The case of d = 1 is called projectivization formula and proved in [Kuz07, Theorem 5.5], [JL,Theorem 3.4], [Todb, Theorem 4.6.11]. The d = 2 case is proved in [Jia,Theorem 6.19].…”

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confidence: 96%