On a birational classification of bundles and reflexive sheaves on surfaces (Q1192069)

It is known that if \(E\) is an ample and spanned vector bundle of rank two on a smooth surface \(X\) with \(c_ 2(E)=1\), then \((X,E)\cong(\mathbb{P}^ 2,{\mathcal O}(1)^{\oplus 2})\). The aim of this paper is to generalize this result to the case in which \(E\) is a reflexive sheaf on a normal surface \(X\), by introducing some new invariants depending also on the singularities of \(E\) and \(X\), and a notion of ``birational equivalence'' between such pairs \((X,E)\). But the definition given in this paper needs some correction. It neglects differences on exceptional sets, but this is too much on many rational surfaces where Pic is generated by \((-1)\)-curves. Unfortunately the reviewer cannot see how it should be corrected.