We study, after Logachev, the geometry of smooth complex Fano threefolds X X with Picard number 1 1 , index 1 1 , and degree 10 10 , and their period map to the moduli space of 10-dimensional principally polarized abelian varieties. We prove that a general such X X has no nontrival automorphisms. By a simple deformation argument and a parameter count, we show that X X is not birational to a quartic double solid, disproving a conjecture of Tyurin. Through a detailed study of the variety of conics contained in X X , a smooth projective irreducible surface of general type with globally generated cotangent bundle, we construct two smooth projective two-dimensional components of the fiber of the period map through a general X X : one is isomorphic to the variety of conics in X X , modulo an involution, another is birationally isomorphic to a moduli space of semistable rank- 2 2 torsion-free sheaves on X X , modulo an involution. The threefolds corresponding to points of these components are obtained from X X via conic and line (birational) transformations. The general fiber of the period map is the disjoint union of an even number of smooth projective surfaces of this type.
We analyze (complex) prime Fano fourfolds of degree 10 and index 2. Mukai gave a complete geometrical description; in particular, most of them are contained in a Grassmannian G(2, 5). They are all unirational and, as in the case of cubic fourfolds, some are rational, as already remarked by Roth in 1949.We show that their middle cohomology is of K3 type and that their period map is dominant, with smooth 4-dimensional fibers, onto a 20-dimensional bounded symmetric period domain of type IV. Following Hassett, we say that such a fourfold is special if it contains a surface whose cohomology class does not come from the Grassmannian G(2, 5). Special fourfolds correspond to a countable union of hypersurfaces (the Noether-Lefschetz locus) in the period domain, labelled by a positive integer d. We describe special fourfolds for some low values of d. We also characterize those integers d for which special fourfolds do exist. . O. Debarre and L. Manivel are part of the project VSHMOD-2009 ANR-09-BLAN-0104-01.1 This means that the lines in P(V 5 ) parametrized by Σ 0 all pass through a fixed point. Since X is smooth, it contains no 3-planes by the Lefschetz theorem, hence Σ 0 is indeed a surface.2 If Σ ⊂ X is a σ-quadric, its span P(V ′ 1 ∧ V 5 ) is contained in the isotropic Grassmannian G ω (2, V 5 ), hence V ′ 1 is the kernel of ω and Σ = Σ 0 .
Abstract. The main result of this paper is that the variety of presentations of a general cubic form f in 6 variables as a sum of 10 cubes is isomorphic to the Fano variety of lines of a cubic 4-fold F , in general different from F = Z(f ).A general K3 surface S of genus 8 determines uniquely a pair of cubic 4-folds: the apolar cubic F (S) and the dual Pfaffian cubic F (S) (or for simplicity F and F ). As Beauville and Donagi have shown, the Fano variety F F of lines on the cubic F is isomorphic to the Hilbert scheme Hilb 2 S of length two subschemes of S. The first main result of this paper is that Hilb 2 S parametrizes the variety V SP (F, 10) of presentations of the cubic form f , with F = Z(f ), as a sum of 10 cubes, which yields an isomorphism between F F and V SP (F, 10). Furthermore, we show that V SP (F, 10) sets up a (6, 10) correspondence between F and F F . The main result follows by a deformation argument.1. Pfaffian and apolar cubic 4-folds associated to K3 surfaces of genus 8 1.1. Let V be a 6-dimensional vector space over C. Fix a basis e 0 , . . . , e 5 for V ; then e i ∧ e j for 0 ≤ i < j ≤ 5 form a basis for the Plücker space ∧ 2 V of 2-dimensional subspaces in V or lines in P 5 = P(V ). We associate to a 2-vector g = i
-O'Grady showed that certain special sextics in P 5 called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.R. -O'Grady a démontré que certaines sextiques spéciales dans P 5 , les sextiques EPW, admettent pour revêtements doubles des variétés symplectiques holomorphes lisses. Nous proposons une nouvelle approche de ces variétés symplectiques, en montrant qu'elles se construisent à partir des schémas de Hilbert de coniques sur des variétés de Fano de dimension quatre et de degré dix. En guise d'application, nous construisons des familles de surfaces lagrangiennes dans ces variétés symplectiques, puis des systèmes intégrables dont les fibres sont des jacobiennes intermédiaires.
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