Resultats sur la conjecture de dualite etrange sur le plan projectif

摘要

Le Potier's Strange Duality' conjecture gives an isomorphism between the space of sections of the determinant bundle on two different moduli spaces of semi-stable sheaves on the complex projective plane P_2. We consider two orthogonal classes c, u in the Grothendieck algebra K(P_2) such that c is of positive rank and u of rank zero, and we call M_c and M_u the moduli spaces of semi-stable sheaves of class c, respectively u on P_2. There exists on M_c (resp. M_u) a determinant bundle D_u (resp. D_c) and the product fibre bundle D_c D_c on the product space M_c M_c has a canonical section σ_(c,u) which provides a linear application D_(c,u): H~0(M_u, D_c)~* → H~0(M_c, D_u). If M_c is not empty, D_(c,u) is conjectured to be an isomorphism. We prove the conjecture in the particular case where c is of rank 2, zero first Chern class and second Chern class c_2(c) ≤ 5, and u is of degree d(u) ≤ 3 and zero Euler-Poincare characteristic. In addition we give the generating series P(t) = ∑_(k≥0)t~kh~0(M_c, D_u~((direct X)k)) for c_2(c) = 3, c_2(c) = 4, d(u) = 1, for the particular classes c and u considered above.