Twisted cubics on cubic fourfolds

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

doi:10.48550/arxiv.1305.0178

Cited by 10 publications

(26 citation statements)

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“…The polarization has divisibility 2 and degree either 6n if 3 does not divide n, or 2 3 n otherwise. When a = 1 and b = 1, this is nothing but the family of Fano varieties of lines in the cubic fourfold from [BD85] (with the Plücker polarization of degree 6), while when a = 2 and b = 1, we find the family of polarized eightfold from [LLSvS17] (with the Plücker polarization of degree 2). For the proof of these two examples, we refer to [LPZ18] (see also [LLMS18]).…”

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confidence: 87%

“…Let S be the moduli space of cubic fourfolds. If we choose v = λ 1 + λ 2 in Theorem 29.4, then by [LPZ18] S 0 = S, and M (v) is the relative Fano variety of lines over S. For v = 2λ 1 + λ 2 , still by [LPZ18] (see also [LLMS18]), we have S 0 ⊂ S is the complement of cubics containing a plane, and M 0 (v) is the family of irreducible holomorphic symplectic eightfolds constructed by Lehn, Lehn, Sorger and van Straten [LLSvS17]. Finally, for v = 2λ 1 , we expect an algebraic construction of a 20-dimensional family of singular 10-dimensional O'Grady spaces whose resolution should be birational to the construction in [Voi18].…”

Section: Donaldson-thomas Invariantsmentioning

confidence: 99%

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Stability conditions in families

Bayer,

Lahoz,

Macrì

et al. 2019

Preprint

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We develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich-Polishchuk, Kuznetsov, Lieblich, and Piyaratne-Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai's theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington-Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson-Thomas invariants counting Bridgeland stable objects on Calabi-Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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confidence: 87%

Section: Donaldson-thomas Invariantsmentioning

confidence: 99%

Stability conditions in families

Bayer,

Lahoz,

Macrì

et al. 2019

Preprint

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“…Some interesting examples of such moduli spaces have already been studied in detail since the first version of this article appeared. For example, in [LPZ18], the Fano variety of lines on a cubic fourfold is described as a moduli space of stable objects in general (without the assumption on (−2)-classes appearing in the proof of Theorem A.1), as well as the 8-dimensional hyperkähler variety associated to cubic fourfolds via the Hilbert scheme of twisted cubics [LLSvS17] (thus extending [LLMS18], which only holds for a very general cubic fourfold; this construction also behaves well in family over the moduli space of cubic fourfolds, thus including the results in [Ouc17] when the cubic fourfold contains a plane).…”

Section: Introductionmentioning

confidence: 99%

Stability conditions on Kuznetsov components

Bayer¹,

Lahoz²,

Macrì³

et al. 2017

Preprint

199

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We introduce a general method to induce Bridgeland stability conditions on semiorthogonal decompositions. In particular, we prove the existence of Bridgeland stability conditions on the Kuznetsov component of the derived category of many Fano threefolds (including all but one deformation type of Picard rank one), and of cubic fourfolds. As an application, in the appendix, written jointly with Xiaolei Zhao, we give a variant of the proof of the Torelli theorem for cubic fourfolds by Huybrechts and Rennemo.

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“…the blow-up of the diagonal in the symmetric product of a K3 surface), see [19]. By varying A ∈ LG( 3 V ) 0 one gets a locally versal family of HK varieties -one of the five known such families in dimensions greater than 2, see [3,4,9,10,11] for the construction of the other families. The complement of LG( 3 V ) 0 in LG( 3 V ) is the union of two prime divisors, Σ and ∆; the former consists of those A containing a non-zero decomposable tri-vector, the latter is defined in Subsection 1.5.…”

Section: Introductionmentioning

confidence: 99%

Periods of double EPW-sextics

O’Grady

2015

Math. Z.

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We study the period map for double EPW-sextics, which are varieties making up a locally versal family of polarized hyperkaehler fourfolds. Double EPW-sextics are parametrized by Lagrangian subspaces of the third wedge-product of a 6-dimensional complex vector space. We prove that Lagrangians in the indeterminacy locus of the period map contain\ud a decomposable 3-vector enjoying very special properties. In addition we prove a result about periods of double EPW-sextics which are either smooth or have isolated singular points with projectivized tangent cone isomorphic to the incidence variety in P2 × (P2)∨

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Twisted cubics on cubic fourfolds

Cited by 10 publications

References 0 publications

Stability conditions in families

Stability conditions in families

Stability conditions on Kuznetsov components

Periods of double EPW-sextics

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