Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

Feyzbakhsh, Soheyla; Pertusi, Laura

doi:10.48550/arxiv.2109.13549

Cited by 5 publications

(8 citation statements)

References 25 publications

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“…Something is known in the rank-two case, cf. Remark 4.5, and very recently the irreducibility for all ranks has been proved in the case of the cubic threefold in [FP,Thm. 1.4].…”

Section: Introductionmentioning

confidence: 93%

Ulrich bundles on Del Pezzo threefolds

Ciliberto¹,

Flamini²,

Knutsen³

2022

Preprint

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We prove that for any r 2 the moduli space of stable Ulrich bundles of rank r and determinant OX (r) on any smooth Fano threefold X of index two is smooth of dimension r 2 + 1 and that the same holds true for even r when the index is four, in which case no odd-rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties. A preliminary lemmaThis short section is devoted to the proof of the following: Lemma 2.1. Let X ⊂ P m be a smooth, irreducible, projective variety with deg(X) > 1 and let E be an Ulrich bundle on X. Then h 0 (E * ) = 0.Proof. Assume by contradiction that h 0 (E * ) > 0 and pick a non-zero section s * ∈ H 0 (E * ). Since h 0 (E) = deg(X) rk(E) deg(X) > 1 by [Be1, (3.1)], we have

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“…Something is known in the rank-two case, cf. Remark 4.5, and very recently the irreducibility for all ranks has been proved in the case of the cubic threefold in [FP,Thm. 1.4].…”

Section: Introductionmentioning

confidence: 93%

Ulrich bundles on Del Pezzo threefolds

Ciliberto¹,

Flamini²,

Knutsen³

2022

Preprint

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“…(1) By [114,Proposition 5.7] the stability conditions σ(s, q) on Ku(Y d ) are Serre-invariant for every 1 ≤ d ≤ 5. ( 2) By [61,Theorem 4.25] (see also [48,Remark 3.7]) there is a unique GL Note that by [84] every equivalence is of Fourier-Mukai type in this case. (4) The Categorical Torelli theorem holds for Y 4 by [27], and for Y 5 since it is unique up to isomorphism by [60].…”

Section: Cubic Threefolds and Beyondmentioning

confidence: 99%

“…This is applied to cubic threefolds in Section 5.3 to construct stability conditions on the associated Kuznetsov component which are Serre-invariant. In Section 5.4 we explain some applications of this result on Serre-invariant stability conditions to the study of the geometry of moduli spaces and to give an alternative proof of the Categorical Torelli theorem; the main references are [7,48,114]. In Section 5.5 we recall the state of art about these questions on Serre-invariant stability conditions and Categorical Torelli theorem for the Kuznetsov component of prime Fano threefolds of index 2 and 1.…”

Section: Introductionmentioning

confidence: 99%

Categorical Torelli theorems: results and open problems

Pertusi¹,

Stellari²

2022

Preprint

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We survey some recent results concerning the so called Categorical Torelli problem. This is to say how one can reconstruct a smooth projective variety up to isomorphism, by using the homological properties of special admissible subcategories of the bounded derived category of coherent sheaves of such a variety. The focus is on Enriques surfaces, prime Fano threefolds and cubic fourfolds.Contents 24 6. Cubic fourfolds 40 References 47

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“…There is however only a short list of varieties for which ACM or Ulrich bundles are completely classified. For works regarding ACM and Ulrich bundles on Fano threefolds, mostly of low ranks, we refer to [AC,Be1,Be2,Be3,Be4,BF1,BF2,BF3,CH,CKL,CM,CFaM1,CFaM2,CFaM3,CFiM,CFK,FP,LMS,LP,MT].…”

Section: Introductionmentioning

confidence: 99%

Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

Ciliberto¹,

Flamini²,

Knutsen³

2022

Preprint

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Let X be any smooth prime Fano threefold of degree 2g − 2 in P g+1 , with g ∈ {3, . . . , 10, 12}. We prove that for any integer d satisfying g+3 2 d g + 3 the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when (g, d) = (4, 3) and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank-two slope-stable ACM bundles F d on X such that det(F d ) = OX (1), c2(F d )•OX (1) = d and h 0 (F d (−1)) = 0 is nonempty and has a component of dimension 2d − g − 2, which is furthermore reduced except for the case when (g, d) = (4, 3) and X is contained in a singular quadric. This completes the classification of rank-two ACM bundles on prime Fano threefolds. Secondly, we prove that for every h ∈ Z + the moduli space of stable Ulrich bundles E of rank 2h and determinant OX (3h) on X is nonempty and has a reduced component of dimension h 2 (g + 3) + 1; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.

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Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

Cited by 5 publications

References 25 publications

Ulrich bundles on Del Pezzo threefolds

Ulrich bundles on Del Pezzo threefolds

Categorical Torelli theorems: results and open problems

Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

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