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 ∈{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 (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 threefolds. Secondly, we prove that for every h ∈ℤ^+ the moduli space of stable Ulrich bundles ℰ of rank 2 h 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 .