Rank two weak Fano bundles on del Pezzo threefolds of degree five

This paper classifies rank two vector bundles on a del Pezzo threefold $X$ of degree five whose projectivizations are weak Fano. This classification is then used to determine properties of the moduli spaces of such vector bundles on $X$, and we determine precisely when the moduli spaces are smooth, irreducible, and fine. We also prove that such a bundle on a del Pezzo threefold of degree one or two splits, and as result give a classification of weak Fano bundles of rank two on a del Pezzo threefold of Picard rank one.