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

Abstract

Let X be any smooth prime Fano threefold of degree 2g − 2 in P g+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 ) = OX (1), c2(F d )•OX (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 threefolds. Secondly, we prove that for every h ∈ Z + the moduli space of stable Ulrich bundles E of rank 2h and determinant OX (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.