On the moduli space of the Schwarzenberger bundles

Abstract

For any odd n, we prove that the coherent sheaf F A on P n C , defined as the cokernel of an injective map f : O ○+2 Pn → O Pn (1) ○+(n+2) , is Mumford-Takemoto stable if and only if the map f is stable, when considered as a point of the projective space P(Hom(O Pn (-1) ⊗2 ,O ⊗(n+2) Pn )*) under the action of the reductive group SL(2) × SL(n + 2). This proves a particular case of a conjecture of J.-M.Drezet and it implies that a component of the Maruyama scheme of the semi-stable sheaves on P n of rank n and Chern polynomial (1 + t) n+2 is isomorphic to the Kronecher moduli N(n + 1, 2, n + 2), for any odd n. In particular, such scheme defines a smooth minimal compactification of the moduli space of the rational normal curves in P n , that generalizes the construction defined by G. Ellinsgrud, R. Piene and S. Stromme in the case n = 3.