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 2g-2 in {\mathbb P}^{g+1}, with g \in \{3,\ldots ,10,12\}. We prove that for any integer d satisfying \left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant 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 {\mathcal F}_d on X such that \det ({\mathcal F}_d)={\mathcal O}_X(1) , c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d and h^0({\mathcal 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 \in {\mathbb Z}^+ the moduli space of stable Ulrich bundles {\mathcal E} of rank 2h and determinant {\mathcal 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.