### Varieities with low dimensional dual variety.

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We study threefolds $X\subset {\mathbb{P}}^{r}$ having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings...

In this paper we classify rank two Fano bundles $\mathcal{E}$ on Fano manifolds satisfying ${H}^{2}(X,\mathbb{Z})\cong {H}^{4}(X,\mathbb{Z})\cong \mathbb{Z}$. The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $\mathbb{P}\left(\mathcal{E}\right)$, that allows us to obtain the cohomological invariants of $X$ and $\mathcal{E}$. As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

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