Rank 2 vector bundles over \(\mathbb P^2(\mathbb C)\) whose sections vanish on points in general position
A \(0\)-dimensional subscheme \(W\) of \(\mathbb{P}^2_{\mathbb C}\) is said to be in \textit{very general position} if each subscheme \(W' \subseteq W\) imposes \(\mathrm{lenght}(W')\) conditions to any complete linear system of plane curves. The paper under review deals with the problem of classification of rank two vector bundles \(E\) such that \(h^0(\mathbb{P}^2,E) \geq 2\) and such that the general section is vanishing in a \(0\)-dimensional scheme in very general position (ZGP bundles). The authors are able to find all the possible values of the Chern classes of such \(E\)'s and subsequently their stability (with some exceptions). The complete classification according to their second Chern class is presented (last part of Sect. 4). Afterwards they partially ask the problem of fitting a ZGP bundle \(E\) in a particular exact sequence which leads to constructions of several new examples of threefolds \(\mathbb{P}(E)\) of low degree (\(11\) in \(\mathbb{P}^{10}\), for instance, see Theorem 9).
scientific article; zbMATH DE number 6484513