Twisted cubics on cubic fourfolds

Lehn, Christian; Lehn, Manfred; Sorger, Christoph; Straten, Duco van

doi:10.1515/crelle-2014-0144

Cited by 87 publications

(125 citation statements)

References 29 publications

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“…This allows us to concentrate on the problem of constructing a smooth compactification of J U1 which is flat over B. The fact that the variety we construct is irreducible holomorphic symplectic (or hyper-Kähler) makes use of [42]. Indeed, the intermediate Jacobian fibration contains a divisor which is birationally a P 1 -bundle on the hyper-Kähler 8-fold recently constructed in [42].…”

Section: The Hyper-kähler Structurementioning

confidence: 99%

“…For instance [13], one of the key papers in the field, shows that the Fano variety of lines of a cubic X is a deformation of a Hilb 2 (K3). More recently, [42] constructs a HK manifold from the variety of cubic rational curves in X, which is then shown in [1] to be deformation equivalent to a Hilb 4 (K3).…”

Section: Introductionmentioning

confidence: 99%

“…The variety J contains the Zariski closure of the Theta divisor Θ⊂J which is birational to a P 1 -bundle over the Lehn-Lehn-Sorger-van Straten variety F 3 (X). Indeed, recall from [42] that F 3 (X) parameterizes birationally nets |D| of rational cubic curves on cubic surfaces S ⊂X. Consider the P 1 -bundle P!F 3 (X) having as fiber over (S, D)…”

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confidence: 99%

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A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

2017

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Section: The Hyper-kähler Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

2017

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“…This morphism induces a fibration M 3 (W ) → Z ′ (W ), which turns out to be a P 2 -fiber bundle. With some further work, the authors of [LLSvS17] prove that the variety Z ′ (W ) is also smooth and projective of dimension 8.…”

Section: Moduli Spacesmentioning

confidence: 99%

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

Macrì

Stellari

2019

Lecture Notes of the Unione Matematica Italiana

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We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds.These notes originated from the lecture series by the first author at the school on BirationalFrom the homological point of view, it was observed by Kuznetsov [Kuz10] that the derived category D b (W ) of a cubic fourfold W contains an admissible subcategory K u(W ) which is the right orthogonal of the category generated by the three line bundles O W , O W (H) and O W (2H). We will refer to K u(W ) as the Kuznetsov component of W . The category K u(W ) has the same homological properties of D b (S), for S a K3 surface: it is an indecomposable category with Serre functor which is the shift by 2 and the same Hochschild homology as D b (S). But, for W very general, there cannot be a K3 surface S with an equivalence K u(W ) ∼ = D b (S). This is the reason why we should think of Kuznetsov components as non-commutative K3 surfaces.The study of non-commutative varieties was started more than thirty years ago. Artin and Zhang [AZ84] investigated the case of non-commutative projective spaces (see also the book in preparation [Yek18]). In these notes we will follow closely the approach developed by Kuznetsov [Kuz10] and Huybrechts [Huy17]. One important feature is that the Kuznetzov component comes with a naturally associated lattice H(K u(W ), Z) with a weight-2 Hodge Non-commutative Calabi-Yau varietiesIn this section, we follow closely the presentation and main results in [Kuz15]; foundational references are also [BO95, BvdB03, Kuz07, Kuz14, Perr18a]. We start with a very short review on semiorthogonal decompositions and exceptional collections in Section 2.1. We then introduce the notion of non-commutative variety in Section 2.2 and study basic facts about Serre functors and Hochschild (co)homology. Finally, in Section 2.3 we sketch the

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“…In this section we deal only with aCM curves and we also assume that the surface S has only ADE singularities. In this case every aCM curve belongs to a two-dimensional linear system with smooth general member, just as in the case of smooth S [LLSvS,Theorem 2.1]. Moreover, these linear systems are in one-to-one correspondence with the determinantal representations of S. Let us explain this in detail.…”

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confidence: 99%

On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold

Shinder¹,

Soldatenkov²

2017

Kyoto J. Math.

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Abstract. In this note we make a few remarks about the geometry of the holomorphic symplectic manifold Z constructed in [LLSvS] as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold Y ⊂ P 5 .We show that Z is birational to a component of the moduli space of stable sheaves in the Calabi-Yau subcategory of the derived category of Y . Using this description we deduce that the twisted cubics

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Twisted cubics on cubic fourfolds

Abstract: ABSTRACT. We construct a new twenty-dimensional family of projective eight-dimen-

Cited by 87 publications

References 29 publications

A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces

On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold

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