Strange duality on rational surfaces (Q2293739)

In the paper under review, the author proves some cases of Le Potier's strange duality conjecture for rational surfaces \(X\). To state it, let \(c,u\) be two elements of the Grothendieck group \(K(X)\) of coherent sheaves on \(X\) and let us suppose that \(c\) and \(u\) are orthogonal under the quadratic form defined by the Euler characteristic. Let \(H\) be an ample line bundle with \(H.K_X<0\) and let us consider \(M_X^H(c)\) (resp. \(M_X^H(u)\)) the moduli space of H-semistable sheaves of class c (resp. u). Under certain technical conditions, it is possible to define a line bundle \(\lambda_c(u)\) (resp. \(\lambda_u(c)\)) on \(M_X^H(c)\) (resp. on \(M_X^H(u)\)) and a canonical map \[ SD_{c,u} :H^0(M^H_X(c),\lambda_c(u))^{\vee}\rightarrow H^0(M^H_X(u),\lambda_u(c)) \] The strange duality conjecture states that this map is an isomorphism. In this paper, the conjecture is proved when \(M^H_X(c)\) is the moduli space of rank \(2\) sheaves with trivial first Chern class and second Chern class \(2\) and \(M^H_X(u)\) is the moduli space of \(1\)-dimensional sheaves with determinant a suitable line bundle \(L\) and Euler characteristic \(0\). For instance, the conjecture holds for any ample line bundle \(L\) on Hirzebruch surfaces \(\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(e))\) when \(e\neq 1\).