On the Buchsbaum index of rank two vector bundles on \({\mathbb P}^3\) (Q2809907)

For a rank \(2\) vector bundle \(E\) on \({\mathbb P}^3\) the Buchsbaum index is \(b(E):= \text{min} \{ \ell | {\mathfrak m}^\ell H^1_*(E)=0 \}\), \(\mathfrak m\) being the irrelevant ideal. In this paper one classifies rank \(2\) vector bundles on \({\mathbb P}^3\) with \(b(E)=3\). The main result is Theorem 1.2. in which one shows that \(b(E)=3\) implies that ``\(E\) is stable and:NEWLINENEWLINE (i) if \(c_1(E)=0\), \(E\) is an instanton with \(3 \leq c_2(E) \leq 5\). Moreover for any \(3 \leq c_2 \leq 5\), there exists an instanton, \(E\), with \(c_2(E)=c_2\) and \(b(E)=3\);NEWLINENEWLINE (ii) if \(c_1=-1\), then \(c_2=2\). Every stable bundle \(E\) with \(c_1=-1\), \(c_2=2\) has \(b(E)=3\).''NEWLINENEWLINEThe connection with the existing literature is described in the paper, including cases \(b(E)=1\) and \(b(E)=2\), studied by \textit{P. Ellia} and \textit{A. Sarti} [Lect. Notes Pure Appl. Math. 206, 81--92 (1999; Zbl 0960.14026)].