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We deal with instanton bundles on the product {\mathbb{P}^{1}\times\mathbb{P}^{2}} and the blow up of {\mathbb{P}^{3}} along a line. We give an explicit construction leading to instanton bundles. Moreover, we also show that they correspond to smooth points of a unique irreducible component of their moduli space.

In this paper we deal with a particular class of rank two vector bundles (instanton bundles) on the Fano threefold of index one F := F 1 × P 1 . We show that every instanton bundle on F can be described as the cohomology of a monad whose terms are free sheaves. Furthermore we prove the existence of instanton bundles for any admissible second Chern class and we construct a nice component of the moduli space where they sit. Finally we show that minimal instanton bundles (i.e. with the least possible degree of the second Chern class) are aCM and we describe their moduli space.

We prove that every Ulrich bundle on the Veronese surface has a resolution in terms of twists of the trivial bundle over P 2 . Using this classification, we prove existence results for stable Ulrich bundles over P k with respect to an arbitrary polarization dH.

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