The moduli spaces of some rank-2 stable vector bundles over algebraic \(K\)3-surfaces (Q1190845)

The author proves that certain moduli spaces of stable rank-two vector- bundles on a \(K3\)-surface \(S\) are birational to symmetric products of \(S\). The conditions on the vector bundles \(F\) whose moduli are being considered, are the following: 1. \(\text{det} F\cong{\mathcal O}_ S\), and 2. \(c_ 2(F)=H^ 2n^ 2+3\) for some positive integer \(n\), where \(H\) is the polarization of \(S\). The result is proved as follows. By Riemann-Roch and a dimension count, if \(F\) is the generic \(H\)-stable rank-two vector bundle as above, then there exists a unique exact sequence \[ 0\to{\mathcal O}_ S\to F(nH)\to I_ Z(2nH)\to 0, \] where \(Z\) is a zero-dimensional subscheme of \(S\) of length \(2H^ 2n^ 2+3\). Conversely, if \(Z\) is a generic such subscheme, there exists a unique exact sequence as above. This gives the result.