Determinant surfaces of rank 2 bundles on \(\mathbb{P}^ 3\) (Q1346791)

The aim of this paper is to study the relationship between stable vector bundles of rank two on projective three space and their determinant surface determined by two sections of the bundle. A general surface has Picard number \(\rho (S) = 1\). A smooth surface \(S\) is not general because its Picard number is at least two by the following result of the author: A smooth surface \(S \subset \mathbb{P}^3\) occurs as a determinant of a rank two vector bundle \({\mathcal E}\) on \(\mathbb{P}^3\) if and only if \(S\) has a surjective morphism onto \(\mathbb{P}^1\). In the following the author estimates \(\rho (S)\). There is an estimate of \(\rho (S)\) from below in terms of the behaviour of \({\mathcal E}\) under the restriction to lines and planes. In particular, defining jumping planes there is a sufficient condition for \(S\) to have Picard number \(\rho (S) \geq 3\).