Rationality of moduli spaces of torsion free sheaves over rational surfaces (Q1911177)

Let \(S\) be a smooth rational ruled surface with \(F\) a general fibre. Let \(H\) be an ample divisor with \((K_S+F)\cdot H<0\). Then the author shows that in most cases, the moduli space \(M_H^s(r,c_1,c_2)\) of \(H\)-Gieseker semistable sheaves of rank \(r\), first Chern class \(c_1\) and \(c_1\cdot F=0\) and given (large enough, with some additional condition) \(c_2\), is rational. \textit{A. Hirschowitz} and \textit{Y. Laszlo} [Math. Ann. 297, No. 1, 85-102 (1993; Zbl 0808.14007)] introduced the notion of priority sheaves over \(\mathbb{P}^2\) and thus proved that the moduli space as above is irreducible. \textit{C. Walter} generalised this notion to arbitrary birationally ruled rational surfaces, in his Nice preprint (1993), which the reviewer has not seen in print. A torsion free sheaf \(E\) on a rational ruled surface \(S\) is called prioritary, if \(\Hom(E,E(K_S+ F))=0\). (In the Hirschowitz-Laszlo definition, \(\text{Ext}^2 (E,E(-1))=0\), because they are over \(\mathbb{P}^2\).) Walter proved that the appropriate moduli stack is irreducible and smooth. Now, the condition \((K_S+F)\cdot H<0\) becomes relevant, since under that condition, every \(H\)-Gieseker semistable sheaf is prioritary. It is also true that such an \(H\) exists.