Vector bundles of Grassmann type and configuration type of rank 2 on an algebraic surface (Q805690)

\textit{N. Sasakura} [see RIMS Kokyuroku Report 634 (Kyoto Univ. 1987; Zbl 0695.32002); p. 407-513] had given a method to construct reflexive sheaves on complex spaces called sheaves of type \((G)\) or type \((C)\). The former are constructed by using sections of a line bundle while the latter by using configurations of divisors. The problem considered here is to determine Chern classes of rank two vector bundles of these types on a nonsingular projective surface S over \({\mathbb{C}}\). The author succeeds in getting a partial but useful solution to this problem. As an application he shows that given L and \(C_ 2\) there exists an integer m and a vector bundle E (of rank two) of type \((G)\) such that \(C_ 1(E(m))=L\), \(C_ 2(E(m))=C_ 2\) [compare: \textit{R. L. E. Schwarzenberger}, Proc. Lond. Math. Soc., III. Ser. 11, 623-640 (1961; Zbl 0212.260)]. Similar results are proved for rank two vector bundles of type \((C)\) for \(S={\mathbb{P}}^ 2\). An interesting corollary is that every stable rank two vector bundle on \({\mathbb{P}}^ 2\) is a deformation of a vector bundle of type \((C)\) twisted by \({\mathcal O}(n)\).