Numerical invariants for bundles on blow-ups (Q2750881)

The authors consider rank 2 vector bundles \(E\) with vanishing first Chern class on the blow-up \(\pi:X\to \mathbb C^2\) 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 isomorphic to \({\mathcal O} (j)\oplus {\mathcal O} (-j)\). Using a canonical form of the transition matrix of such bundles, the set \(M_j\) can be described as a quotient of \(\mathbb C^N\) by an equivalence relation [\textit{E. Gasparim}, J. Algebra 199, 581--590 (1998; Zbl 0902.14030)].NEWLINENEWLINEEquipped with the quotient topology, \(M_j\) is not Hausdorff. By fixing two numerical invariants of the bundle \(E\), this space is decomposed into subspaces which turn out to be homeomorphic to open sets in \(\mathbb C\mathbb P^n\) with \(0\leq n \leq 2j-3\). The two numerical invariants are the length of \((\pi_\ast E)^{\vee\vee}/\pi_\ast E\) and the length of \(R^1\pi_\ast E\). The authors describe a procedure which calculates these invariants in terms of the transition matrix.