On the generation of certain bundles over \(\mathbb{P}^ 3\)
In a 1988 preprint, \textit{Ph. Ellia} and \textit{A. Hirschowitz} [see also J. Algebraic Geom. 1, No. 4, 531-547 (1993)] proposed the following principle: If a locally free sheaf \(F\) of rank \(r\) on a smooth projective variety of dimension \(n\) is generated by its global sections, then it should be \(h^ 0F\geq r+n\). The converse should be true if \(F\) is general enough. From this principle it follows the conjecture: Fix \(n\geq 2\). For \(a,b,d\) positive integers such that \(d\geq 1\), \(a\geq 1\) and \(\bigl({d+n\choose n}- 1\bigr)b\geq\bigl({d+n+1\choose n}-1\bigr)a+n\) the kernel \(E^ n_{a,b}(d)\) of the generic morphism from \(b{\mathcal O}_{\mathbb{P}^ 3}(d)\) to \(a{\mathcal O}_{\mathbb{P}^ 3}(d+1)\) is generated by its global sections. The case \(d=1\) was discussed by Ph. Ellia and A. Hirschowitz. In this paper, the author considers the case \(n=3\) and \(d=2\).
locally free sheaf generated by its global sections
scientific article; zbMATH DE number 126358