Ample divisors on fine moduli spaces on the projective plane (Q789456)

Let \(c_ 1\) and \(c_ 2\) be two integers with discriminant \(D=c_ 1\!^ 2-4c_ 2\) satisfying \(D<0\), \(D\neq -4\), and \(D\not\equiv 0(mod 8),\) and let \(M(c_ 1,c_ 2)\) be the fine moduli space for stable rank- 2 coherent sheaves on \({\mathbb{P}}^ 2\) with the given Chern classes. We prove that the Picard group of \(M(c_ 1,c_ 2)\) is free on two generators, and the geometric significance of these is given. The ample cone is described in terms of the generators. Furthermore, the divisor class of the non-locally free sheaves is computed, as well as the canonical class.