On the Barth morphism (Q2757193)

Let \(F\) be a rank-2 semi-stable sheaf on the projective plane, with Chern classes \(c_1=0\), \(c_2=n\). The curve \(\beta_F\) of jumping lines of \(F\), in the dual projective plane, has degree \(n\). Let \(M_n\) be the moduli space of equivalence classes of semi-stable sheaves of rank 2 and Chem 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 \(\beta:M_n \to{\mathcal C}_n\) associates the point \(\beta_F\) to the class of the sheaf \(F\). 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 \({\mathcal C}_n\); 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.