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

Let X be any smooth prime Fano threefold of degree 2g-2 in Pg+1, with g ? {3, ... , 10, 12}. We prove that for any integer d satisfying [ g+3/2 ] = d = g +3 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 (g, d) = (4, 3) and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank-two slope-stable ACM bundles F-d on X such that det(F-d) = O-X(1), c(2)(F-d) . O-X(1) = d and h(0)(F-d(-1)) = 0 is nonempty and has a component of dimension 2d - g - 2, which is furthermore reduced except for the case when (g, d) = (4, 3) and X is contained in a singular quadric. This completes the classification of rank-two ACM bundles on prime Fano three folds. Secondly, we prove that for every h ? Z(+) the moduli space of stable Ulrich bundles e of rank 2h and determinant O-X (3h) on X is nonempty and has a reduced component of dimension h(2)(g + 3) + 1; 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.