A note on a paper of J.-M. Drezet on the local factoriality of some moduli spaces (Q2785299)

Let \(X\) be a smooth complex projective surface and \(H\) a very ample divisor on \(X\). Then, with respect to the polarization \(H\) of \(X\), there is a coarse moduli space \(\overline {M_H} (r,c_1,c_2)\) for the semi-stable sheaves of rank \(r\) and given Chern classes \(c_1\) and \(c_2\) over \(X\). For a rational surface \(X\) with canonical bundle \(K_X\) and a polarization divisor \(H\) satisfying \(K_X \cdot H<0\), \textit{J.-M. Drézet} [J. Reine Angew. Math. 413, 99-126 (1991; Zbl 0711.14029)] had studied the property of factoriality for the local rings at the closed points of the corresponding moduli spaces. Drézet's result states that the closed points of \(\overline M_H(r,c_1, c_2)\) can be distinguished by two types (type I and type II), where the moduli space is never locally factorial at points of type II. Moreover, he gave sufficient conditions for the local factoriality at points of type I.NEWLINENEWLINENEWLINEIn the present note, the author takes up the problem of local factoriality for the moduli spaces \(\overline M_H (r,c_1, c_2)\) and presents some further-going results. Actually, if \(X\) is a smooth rational surface non-isomorphic to \(\mathbb{P}^2\), then there is a morphism \(f:X\to \mathbb{P}^1\) with generic fibre \(\mathbb{P}^1\). The author's main result states that if \((K_X+F) \cdot H<0\) for a fibre \(F\) of \(f\), then \(\overline M_H(r,c_1, c_2)\) is locally factorial at all closed points of type I. -- The proof is based upon a subtle analysis of prioritary sheaves (in the sense of Hirschowitz-Lászlo) and the well-known deformation theory of Drézet and Le Potier for semi-stable sheaves.