Sur le morphisme de Barth

Abstract

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.