Even and odd instanton bundles on Fano threefolds of Picard number one

Abstract

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 ).

Cohomological characterisation of monads

“…The very recent works on instanton bundles on Fano threefolds by Faenzi and Kuznetsov show that, as in the case of the projective space, instanton bundles on V 5 have a description in terms of monads: a k ‐instanton bundle on V 5 , , can be defined as the cohomology of a monad of the form So the above corollary gives also a cohomological characterisation of instanton bundles on V 5 . The existence of k ‐instanton bundles on Fano threefolds of index 2 for is proved by Faenzi (, Theorem 3.1).…”

“…The very recent works on instanton bundles on Fano threefolds by Faenzi [8] and Kuznetsov [19] show that, as in the case of the projective space, instanton bundles on V 5 have a description in terms of monads: a k-instanton bundle on V 5 , k ≥ 2, can be defined as the cohomology of a monad of the form…”

“…They furthermore proved a cohomological characterisation of sheaves on that are the cohomology of a monad of the form and a cohomological characterisation of sheaves on that are the cohomology of a monad Both results were proved by means of the Beilinson spectral sequence. Instanton bundles on Fano threefolds were recently defined and studied in and . The authors proved that in the case of the Fano threefold V 5 of index 2 and degree 5 these bundles can also be defined as the cohomology of certain monads.…”

“…Both results were proved by means of the Beilinson spectral sequence. Instanton bundles on Fano threefolds were recently defined and studied in [8] and [19]. The authors proved that in the case of the Fano threefold V 5 of index 2 and degree 5 these bundles can also be defined as the cohomology of certain monads.…”

Rank two aCM bundles on the del Pezzo fourfold of degree 6 and its general hyperplane section

“…As pointed out in [15], Notation 3.8, a non-empty family M Φ of initialized, indecomposable, Ulrich bundles of rank 2 is defined inside the moduli space of stable vector bundles of rank 2 on Φ with Chern classes γ 1 = 2η, γ 2 = η 2 2 + 3η 2 1 + 2η 1 η 2 . The elements of M Φ fit into an exact sequence (20) 0…”