2022/06/20 by Ciro Ciliberto, Ciliberto, Ciro, Flaminio Flamini +3 · 1 citation
Mathematics · Social Sciences · Arts and Humanities · #Algebraic Geometry and Number Theory #Vietnamese History and Culture Studies #North African History and Literature
paper · pdf · doi:10.48550/arxiv.2206.09986
Let X be any smooth prime Fano threefold of degree 2g-2 in ℙg+1, with g ∈ \3,…,10,12\. We prove that for any integer d satisfying \lfloor (g+3)/(2) \rfloor ≤ 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 Fd on X such that det(Fd)=OX(1), c2(Fd)⋅ OX(1)=d and h0(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 ∈ 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 h2(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.