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

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 $\left\lfloor \frac{g+3}{2} \right\rfloor \leq d \leq g+3$ 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.