Spazi di moduli di fasci aritmeticamente Cohen-Macaulay su varieta di Fano della serie principale
Résumé
In the first part of the paper we complete the classification of the arithmetical Cohen-Macaulay vector bundles of rank 2 on a smooth prime Fano threefold. In the second part, we study some moduli spaces of these vector bundles, using the decomposition of the derived category of X provided by Kuznetsov, when the genus of X is 7 or 9. This allows to prove that such moduli spaces are birational to Brill-Noether varieties of vector bundles on a smooth projective curve γ. When the second Chern class is low we are able to give a more precise description of the moduli space ofrank-2 semistable sheaves with fixed Chern classes Mx(2, c1, c2). If g = 7, we show that the moduli space Mx(2, 1,6) is isomorphic to a smooth irreducible Brill-Noether variety of dimension 3. Moreover the set of vector bundles contained in Mx(2,0,4) is smooth irreducible of dimension 5. Ifg = 9, we prove that Mx(2, 1, 7) is isomorphic to the blow-up of Pic (γ) where γ is a plane smooth quartic. Ifg = 12, an open set of Mx(2, 1, d) can be described as a quotient with respect to the action of a semisimple group in terms of monads.