Cycles, curves and vector bundles on an algebraic surface
Let \({\mathcal D}\) be an invertible sheaf on a smooth algebraic surface X and let \(\xi\) be a 0-dimensional subscheme of X whose ideal sheaf is \(J_{\xi}\). Set \(V=H^ 1(J_{\xi}({\mathcal D}))\) and let \(K_ X\) be the canonical bundle. Then we have a natural exact sequence \(0\to V\otimes K_ X\to E(\xi,{\mathcal D})\to J_{\xi}({\mathcal D})\to 0\), the extension class of which corresponds to the identity homomorphism of V by Serre duality. There is a simple criterion for this sheaf E(\(\xi\),\({\mathcal D})\) to be locally free. Another method to construct a vector bundle from a smooth curve on X and a complete linear system on C without fixed points is described. These techniques are applied to several problems related to vector bundles on K3-surfaces. In particular, the structure of the moduli spaces of simple vector bundles on K3-surfaces is studied, and Lazarsfeld's results on Brill-Noether curves on K3-surfaces are discussed. (Reviewer's remark: The proof of theorem 4.1 seems to contain a gap, since the moduli space may have many connected components. The argument shows only that one of them is birational to the symmetric product of X. According to S. Mukai, there are actually counter-examples.)
scientific article; zbMATH DE number 4025581