Rank two bundles on the blow-up of \(\mathbb{C}^2\)
The author considers the problem of the classification of rank 2 vector bundles with trivial determinant on the blow-up of \(\mathbb{C}^2\) at the origin. For each integer \(j \geq 0\), let \(M_j\) be the set of isomorphism classes of such bundles whose restriction to the exceptional divisor is of the form \({\mathcal O} (j)\oplus {\mathcal O} (-j)\). By elementary arguments, the author shows that the transition matrix of a bundle in \(M_j\) has a canonical form. Using this result, one gets immediately that \(M_0\) and \(M_1\) consist of one point. The author gives a precise description of \(M_2\). In general, \(M_j\) is the quotient of \(\mathbb{C}^N\) by an equivalence relation and looks, generically, like \(\mathbb{P}^{2j-3} \setminus\) a closed subset of codimension \(\geq 2\). Furthermore, \(M_j\) with the quotient topology is not Hausdorff: For example, \({\mathcal O} (j) \oplus {\mathcal O} (-j)\) is arbitrarily closed to any other bundle in \(M_j\).
scientific article; zbMATH DE number 1117783