Threefolds containing Bordiga surfaces as ample divisors

Abstract:

Let $L$ be an ample line bundle on a smooth complex projective variety $X$ of dimension three such that there exists a smooth member $Z$ of $\vert L\vert$. When the restriction $L_{Z}$ of $L$ to $Z$ is very ample and $(Z,L_{Z})$ is a Bordiga surface, it is proved that there exists an ample vector bundle $\mathcal {E}$ of rank two on $\mathbb {P}^{2}$ with $c_{1}(\mathcal {E}) = 4$ and $3 \leq c_{2}(\mathcal {E}) \leq 10$ such that $(X,L) = (\mathbb {P}_{\mathbb {P}^{2}}(\mathcal {E}),H(\mathcal {E}))$, where $H(\mathcal {E})$ is the tautological line bundle on the projective space bundle $\mathbb {P}_{\mathbb {P}^{2}}(\mathcal {E})$ associated to $\mathcal {E}$.