Sur le morphisme de Barth

Abstract

Let $\mathrm{F}$ be a rank-2 semi-stable sheaf on the projective plane, with Chern classes $c_1=0,c_2=n$. The curve ${\beta }_{\mathrm{F}}$ of jumping lines of $\mathrm{F}$, in the dual projective plane, has degree $n$. Let ${\mathrm{M}}_n$ be the moduli space of equivalence classes of semi-stables sheaves of rank 2 and Chern classes $(0,n)$ on the projective plane and ${\mathcal{C}}_n$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $$\beta :{\mathrm{M}}_n⟶{\mathcal{C}}_n$$ associates the point ${\beta }_{\mathrm{F}}$ to the class of the sheaf $\mathrm{F}$. We prove that this morphism is generically injective for $n\ge 4.$ The image of $\beta$ is a closed subvariety of dimension $4n-3$ of ${\mathcal{C}}_n$; 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.

Plain.tex, 54 pages. Uses diagrams.tex