Sur l'espace de modules des faisceaux semi stables de rang 2, de classes de Chern (0,3) sur ${\Bbb P}_2$

Abstract

The moduli space of semi-stable sheaves of rank-2 and Chern classes (0,3) on the projective plane ℙ 2 , denoted by M=M(0,3), is a smooth irreducible projective variety of dimension 9. In M, the points which come from sheaves which are not locally free give a hypersurface ∂M. In this article we show that all complete surfaces in M must meet the boundary ∂M; in other words, there does not exist a family of vector bundles which is parametrized by a complete surface in M. The essential point of the proof is the construction of a birational morphism Φ:M→, Gr from M to the grassmannian Gr of nets of conics in ℙ 2 ; this allows us to identify M with a blow-up of Gr along a surface. This gives us a precise description of cohomology algebra of M and we use this to determine the fundamental class of a complete surface in M which does not meet ∂M. We then show that this value of the fundamental class can not arise.