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.
Ciliberto, C., Flamini, F., Andreas Leopold Knutsen, (2023). Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds. COLLECTANEA MATHEMATICA, 75(3), 795-822 [10.1007/s13348-023-00413-9].
Elliptic curves, ACM bundles and Ulrich bundles on prime Fano threefolds
Ciro Ciliberto;Flaminio Flamini;
2023-08-20
Abstract
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.File | Dimensione | Formato | |
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