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

Let X be any smooth prime Fano threefold of degree 2g −2 in P^g+1, with g in {3, . . . , 10, 12}. We prove that for any integer d satisfying suitable bounds on g, 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) = OX(1), c2(F_d)·OX(1) = d and h^ (Fd(−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 threefolds. Secondly, we prove that for every h ∈ N, 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 computed 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.