On the Barth morphism

Let F be a rank-2 semi-stable sheaf on the projective plane, with Chem clas ses c(1) = 0, c(2) = n. The Curve beta (F) of jumping lines of F, in the du al projective plane, has degree n. Let M-n be the moduli space of equivalen ce classes of semi-stable sheaves of rank 2 and Chem classes (0, n) on the projective plane and C-n be the projective space of curves of degree n in t he dual projective plane. The Barth morphism beta :M-n --> C-n associates the point beta (F) to the class of the sheaf F. We prove that th is morphism is generically injective for n greater than or equal to 4. The image of beta is a closed subvariety of dimension 4n - 3 of C-n; as a conse quence of our result, the degree of this image is given by the Donaldson nu mber of index 4n - 3 of the projective plane. (C) 2001 Editions scientifiqu es et medicales Elsevier SAS.