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

Abstract

AbstractLet X be any smooth prime Fano threefold of degree $$2g-2$$ 2 g - 2 in $${\mathbb P}^{g+1}$$ P g + 1 , with $$g \in \{3,\ldots ,10,12\}$$ g ∈ { 3 , … , 10 , 12 } . We prove that for any integer d satisfying $$\left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant g+3$$ 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 $$(g,d)=(4,3)$$ ( 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 $${\mathcal F}_d$$ F d on X such that $$\det ({\mathcal F}_d)={\mathcal O}_X(1)$$ det ( F d ) = O X ( 1 ) , $$c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d$$ c 2 ( F d ) · O X ( 1 ) = d and $$h^0({\mathcal F}_d(-1))=0$$ h 0 ( F d ( - 1 ) ) = 0 is nonempty and has a component of dimension $$2d-g-2$$ 2 d - g - 2 , which is furthermore reduced except for the case when $$(g,d)=(4,3)$$ ( 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 $$h \in {\mathbb Z}^+$$ h ∈ Z + the moduli space of stable Ulrich bundles $${\mathcal E}$$ E of rank 2h and determinant $${\mathcal O}_X(3h)$$ O X ( 3 h ) on X is nonempty and has a reduced component of dimension $$h^2(g+3)+1$$ 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.