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Zbl 0998.14007
Le Potier, Joseph; Tikhomirov, Alexander
On the Barth morphism. (Sur le morphisme de Barth.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 34, No. 4, 573-629 (2001). ISSN 0012-9593

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 ${\cal C}_n$ be the projective space of curves of degree $n$ in the dual projective plane. The Barth morphism $\beta:M_n \to{\cal C}_n$ associates the point $\beta_F$ to the class of the sheaf $F$. 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 ${\cal C}_n$; 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.
[V.K.Vedernikov (Moskva)]

MSC 2000:: *14D20 Algebraic moduli problems
14F05 Sheaves, etc.

Keywords: semi-stable sheaf; Chern classes; jumping lines; moduli space; Barth morphism; Donaldson number

Cited in: Zbl 1244.14022 Zbl 1121.14038

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