Resumen de Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds

Ciro Ciliberto, Flaminio Flamini, Andreas Leopold Knutsen

• Let X be any smooth prime Fano threefold of degree in , with . We prove that for any integer d satisfying the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles on X such that , and is nonempty and has a component of dimension , which is furthermore reduced except for the case when and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every the moduli space of stable Ulrich bundles of rank 2h and determinant on X is nonempty and as a reduced component of dimension ; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.