We consider an analogue of the notion of instanton bundle on the projective 3-space, consisting of a class of rank-2 vector bundles defined on smooth Fano threefolds X of Picard number one, having even or odd determinant according to the parity of K X .We first construct a well-behaved irreducible component of their moduli spaces. Then, when the intermediate Jacobian of X is trivial, we look at the associated monads, hyperdeterminants and nets of quadrics. We also study one case where the intermediate Jacobian of X is non-trivial, namely when X is the intersection of two quadrics in 5 , relating instanton bundles on X to vector bundles of higher rank on a the curve of genus 2 naturally associated 2000 Mathematics Subject Classification. Primary 14J60, 14F05. Key words and phrases. Instanton bundles, monads, Fano threefolds, semiorthogonal decomposition, nets of quadrics.The author was partially supported by ANR-09-JCJC-0097-0 INTERLOW and ANR GEOLMI.1 * 3 (−r X ) ≃ 1 , * 2 (−r X ) ≃ 2 . Set U = Hom X ( 2 , 3 ), and note that U ≃ Hom X ( 1 , 2 ).

ABSTRACT. We propose an approach to study logarithmic sheaves n (− log D ) associated with hyperplane arrangements on the projective space n , based on projective duality, direct image functors and vector bundles methods. We focus on freeness of line arrangements having a point with high multiplicity.

We study congruences of lines Xω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n−1. These congruences include the G2-variety for n=6 and the variety of reductions of projected ℙ2×ℙ2 for n=7.\ud We compute the degree of Xω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to Xω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.\ud The residual congruence Y of Xω with respect to a general linear congruence containing Xω is analysed in terms of the quadrics containing the linear span of Xω. We prove that Y is Cohen-Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components

Rank 2 indecomposable arithmetically Cohen-Macaulay bundles E on a nonsingular cubic surface X in P 3 are classified, by means of the possible forms taken by the minimal graded free resolution of E over P 3 . The admissible values of the Chern classes of E are listed and the vanishing locus of a general section of E is studied.Properties of E such as slope (semi)stability and simplicity are investigated; the number of relevant families is computed together with their dimension.

International audienceIn this paper we prove Homological Projective Duality for categorical resolutions of several classes of linear determinantal varieties. By this we mean varieties that are cut out by the minors of a given rank of a m x n matrix of linear forms on a given projective space. As applications, we obtain pairs of derived-equivalent Calabi-Yau manifolds, and address a question by A. Bondal asking whether the derived category of any smooth projective variety can be fully faithfully embedded in the derived category of a smooth Fano variety. Moreover we, discuss the relation between rationality and categorical representability in codimension two for determinantal varieties. (C) 2016 Elsevier Inc. All rights reserved