The Intermediate Jacobian of the Cubic Threefold

Clemens, C. Herbert; Griffiths, Phillip

doi:10.2307/1970801

Cited by 596 publications

(763 citation statements)

References 13 publications

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“…Corollary 1.6 Let X ⊂ P 4 be an irreducible hypersurface covered by the lines of a family of dimension 2 such that Σ is generically reduced. Then the dual varietyX of X is a hypersurface ofP 4 .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Here we have used the condition r ⊂ X. Moreover the homogeneous part of degree 1 in x 2 , x 3 , x 4 of F can be normalized in this way because there is no fixed tangent plane to X along r ( [4]). From P / ∈ Sing(X) it follows that the coefficient of x n−3 1 in the polynomial Ψ is not zero, and we can set Ψ = x n−3…”

Section: Lemma 23 For General H ∈P 4 the Intersection σ ∩ G(1mentioning

confidence: 99%

“…The first ones come from general surfaces contained in G (1,4), while the second ones come from general curves contained in the Hilbert scheme of quadric surfaces in P n . Note that these " quadric bundles" are built by varieties of lower dimension having a higher dimensional family of lines.…”

Section: Introductionmentioning

confidence: 99%

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On threefolds covered by lines

Mezzetti

Portelli

2000

Abh.Math.Semin.Univ.Hambg.

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A classification theorem is given of projective threefolds that are covered by the lines of a two-dimensional family, but not by a higher dimensional family. Precisely, if X is such a threefold, let Σ denote the Fano scheme of lines on X and µ the number of lines contained in X and passing through a general point of X. Assume that Σ is generically reduced. Then µ ≤ 6. Moreover, X is birationally a scroll over a surface (µ = 1), or X is a quadric bundle, or X belongs to a finite list of threefolds of degree at most 6. The smooth varieties of the third type are precisely the Fano threefolds with −K X = 2H X .

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Section: Preliminary Resultsmentioning

confidence: 99%

Section: Lemma 23 For General H ∈P 4 the Intersection σ ∩ G(1mentioning

confidence: 99%

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On threefolds covered by lines

Mezzetti

Portelli

2000

Abh.Math.Semin.Univ.Hambg.

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“…In characteristic 0, the Néron-Severi group NS(F (X)) is indeed torsion-free: this follows from the fact that H 2 (F (X), Z) is a free abelian group ( [CG,Lemmas 9.3 and 9.4]). Standard arguments using smooth and proper base change then imply that in characteristic p > 0, the Néron-Severi group has no prime-to-p torsion.…”

Section: Cubic Threefoldsmentioning

confidence: 99%

Lines on Cubic Hypersurfaces Over Finite Fields

Debarre¹,

Laface²,

Roulleau³

2017

Geometry Over Nonclosed Fields

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Abstract. We show that smooth cubic hypersurfaces of dimension n defined over a finite field F q contain a line defined over F q in each of the following cases:• n = 3 and q ≥ 11;• n = 4, and q = 2 or q ≥ 5;• n ≥ 5. For a smooth cubic threefold X, the variety of lines contained in X is a smooth projective surface F (X) for which the Tate conjecture holds, and we obtain information about the Picard number of F (X) and the 5-dimensional principally polarized Albanese variety A(F (X)).

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“…A series of important results on birational geometry of Fano varieties of index two and higher was obtained by other methods: by the transcendent method of Clemens and Griffiths [3] and its subsequent generalizations (see [2]), and also by means of Kollár's technique [15,16]. For the details, see the introduction to the paper [23], where, in particular, the dramatic story of studying the birational geometry of the Veronese double cone and (not completed to this day) studying of the double space of index two is described.…”

Section: Historical Remarks and Acknowledgementsmentioning

confidence: 99%

Birational geometry of Fano hypersurfaces of index two

Pukhlikov

2015

Math. Ann.

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We prove that every non-trivial structure of a rationally connected fibre space on a generic (in the sense of Zariski topology) hypersurface V of degree M in the (M+1)-dimensional projective space for M ≥ 16 is given by a pencil of hyperplane sections. In particular, the variety V is non-rational and its group of birational selfmaps coincides with the group of biregular automorphisms and for that reason is trivial. The proof is based on the techniques of the method of maximal singularities and inversion of adjunction.

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The Intermediate Jacobian of the Cubic Threefold

Cited by 596 publications

References 13 publications

On threefolds covered by lines

On threefolds covered by lines

Lines on Cubic Hypersurfaces Over Finite Fields

Birational geometry of Fano hypersurfaces of index two

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