A restriction theorem for stable rank two vector bundles on \(\mathbb{P}^3\) (Q2347987)

Let \(E\) be a rank \(2\) vector bundle on \({\mathbb P}^3\), normalized (i.e., twisted with a line bundle such that \(c_1(E)=0 \;\mathrm{ or } -1\)) and stable ( i.e. \(H^0(E)=0\)). By a theorem of \textit{W. Barth} [Math. Ann. 226, 125--150 (1977; Zbl 0332.32021)], the restriction \(E_H\) of \(E\) to a general plane \(H\) is also stable, if \(E\) is not a null-correlation bundle. In this paper one proves that a stable, normalized rank \(2\) vector bundle on \({\mathbb P}^3\) has the property \(h^0(E_H(1)) \leq 2+c_1\), for a general plane \(H\), if \(c_2 \geq 4\). The paper ends with the observation that an indecomposable semistable bundle \(E\) on \({\mathbb P}^3\) with \(c_1(E)=0\), \(h^0(E)\neq 0\), \(h^0(E(-1))=0\) has \(h^0(E_H(1))=3\), for a general plane \(H\).