Lehn’s Formula in Chow and Conjectures of Beauville and Voisin

Maulik, Davesh; Neguţ, Andrei

doi:10.1017/s1474748020000377

Cited by 16 publications

(12 citation statements)

References 17 publications

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“…Note that Γ = ∅ if and only if 0 ≤ n ≤ g − 1; the corollary for the cases n ≥ g follows from the projective bundle formula (Theorem 2. These results are especially interesting in the case when S is a K3 surface [40,45,55]. Note that the map Γ n • Γ n−1 is also given by the correspondence It follows from [15, proof of Proposition 3.2] that codim X ≥1+i ⊂ X ≥ 2i for all i ≥ 1.…”

Section: Q Jiangmentioning

confidence: 96%

On the Chow Theory of Projectivizations

Jiang

2021

J. Inst. Math. Jussieu

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In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$ . In this process, we establish the decomposition of Chow groups for the cases of the Cayley trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds.

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Section: Q Jiangmentioning

confidence: 96%

On the Chow Theory of Projectivizations

Jiang

2021

J. Inst. Math. Jussieu

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“…Proof The first of these equations follows immediately from Corollary A.1 and X = shows the claim. Finally, for mult δ 2 we use the operator e δ of multiplication by δ and its expression in Nakajima operators from [12] in order to compute…”

Section: Proposition 319mentioning

confidence: 99%

The Chow ring of hyperkähler varieties of $$K3^{[2]}$$-type via Lefschetz actions

Kretschmer

2021

Math. Z.

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We propose an explicit conjectural lift of the Neron–Severi Lie algebra of a hyperkähler variety X of $$K3^{[2]}$$ K 3 [ 2 ] -type to the Chow ring of correspondences $$\mathrm{CH}^*(X \times X)$$ CH ∗ ( X × X ) in terms of a canonical lift of the Beauville–Bogomolov class obtained by Markman. We give evidence for this conjecture in the case of the Hilbert scheme of two points of a K3 surface and in the case of the Fano variety of lines of a very general cubic fourfold. Moreover, we show that the Fourier decomposition of the Chow ring of X of Shen and Vial agrees with the eigenspace decomposition of a canonical lift of the cohomological grading operator.

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“…Other applications. The following situations also fit into the framework of "Quot geometry" considered in this paper: Moduli of stable sheaves on surfaces and their Hecke correspondences [Ne17,Ne18,MN19]; Brill-Noether theory of moduli of stable sheaves on K3 surfaces [Mar,AT20], or more generally, Brill-Noether theory of moduli of stable objects in K3 categories [B, BCJ]; The pair of rank two Thaddeus moduli spaces [Tha] (when the parameters are large) with their maps to the moduli of rank 2 vector bundles on curves, considered by [KT]. The results of this paper could be applied verbatim to these situations; we omit the details here in this already long paper.…”

Section: Introductionmentioning

confidence: 99%

“…The geometry of Quot schemes is closely related to correspondence spaces for various moduli spaces: the moduli of stable sheaves on surfaces and their Hecke correspondences studied by Negut ¸,Maulik and Negut ¸[Ne17,Ne18,MN19]; The pair of Thaddeus moduli spaces studied by Thaddeus [Tha], Koseki and Toda [KT]; The moduli spaces from the Brill-Noether theory of moduli of stable sheaves on K3 surfaces studied by Markman [Mar], Addington and Takahashi [AT]; The nested Hilbert schemes of points studied by Gholampour and Thomas [GT1,GT2]; The correspondences in the theory of geometric categorification and Hecke correspondences studied by Cautis, Kamnitzer and Licata [CKL1,CKL2].…”

Section: Introductionmentioning

confidence: 99%

Derived categories of Quot schemes of locally free quotients, I

Jiang¹

2021

Preprint

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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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Lehn’s Formula in Chow and Conjectures of Beauville and Voisin

Cited by 16 publications

References 17 publications

On the Chow Theory of Projectivizations

On the Chow Theory of Projectivizations

The Chow ring of hyperkähler varieties of $$K3^{[2]}$$-type via Lefschetz actions

Derived categories of Quot schemes of locally free quotients, I

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