Moduli of stable sheaves, I

Abstract

Introduction. Let S be a scheme o f finite type over a universally Japanese ring, f: X-4 S b e a smooth, projective, geometrically integral morphisrn and let (9,(1) be an f-very ample invertible sheaf. In this situation, we constructed a moduli scheme M x 1 s (1-!) of stable sheaves with Hilbert polynom inal H in the preceding paper [12] 1). M x i s (//) is locally of finite type and separated over S. And. moreover , M x / s (H) is quasi-projective over S if and only if the family of classes of stable sheaves with Hilbert polynomial H is hounded. A main aim of this article is, under an assumption, to find a natural projective scheme over S which contains M x / s (H) as an open su b sc h e m e. M ore precisely, we shall construct a-moduli scheme" of semi-stable sheaves with Hilbert polynomial H and show th a t the moduli scheme is projective if the family of classes of semi-stable sheaves with Hilbert polynomial H is bounded. As in the case of stable sheaves, our problem is reduced to making a quotient of a suitable open subscheme R o f a Quot-scheme Q by a linear group scheme G. For this purpose, we shall use again the projective bundle Z over a finite union of connected components of the Picard scheme of X /S and the m orphism p of Q to Z which were constructed in § 4 of [ 1 2 ]. In the case of stable sheaves, we had only to show that p maps the points of R corresponding to stable sheaves to stable points of Z. But the case of semi-stable sheaves is more difficult than that because semi-stable points do not have, in general, good functorial properties (see [14] Ch. 1, §5). A way to overcome the difficulty is to show that p(R) is closed in the open subscheme Z s ' of semi-stable points in Z. In fact, when dim X IS < 2 , this was done by C. S.