Smooth Enriques surfaces in \(\mathbb{P}^ 4\) and exceptional bundles (Q1321003)

Assume that \(S\) is a smooth Enriques surface (non-minimal) of degree 10 in \(\mathbb{P}^ 4\), then we prove that on the minimal model of \(S\) there exists a rank 2 vector bundle, stable and exceptional, with \(c^ 2_ 1=14\) and \(c_ 2=4\). Conversely, the existence of such a bundle on a generic Enriques surface \(X\) allows us to give a simple construction of a smooth Enriques surface in \(\mathbb{P}^ 4\).