  Let ${\rm F}$ be a rank-2 semi-stable sheaf on the projective plane, with
Chern classes $c_{1}=0,c_{2}=n$. The curve $\beta_{\rm F}$ of jumping lines of
${\rm F}$, in the dual projective plane, has degree $n$. Let ${\rm M}_{n}$ be
the moduli space of equivalence classes of semi-stables sheaves of rank 2 and
Chern classes $(0,n)$ on the projective plane and ${\cal C}_{n}$ be the
projective space of curves of degree $n$ in the dual projective plane. The
Barth morphism $$\beta: {\rm M}_{n}\longrightarrow{\cal C}_{n}$$ associates the
point $\beta_{\rm F}$ to the class of the sheaf ${\rm F}$. We prove that this
morphism is generically injective for $n\geq 4.$ The image of $\beta$ is a
closed subvariety of dimension $4n-3$ of ${\cal C}_{n}$; as a consequence of
our result, the degree of this image is given by the Donaldson number of index
$4n-3$ of the projective plane.
