Projectivity of the Space of Divisors on a Normal Compact Complex Space

Abstract

For any complex space we shall denote by Dx the Douady space of compact complex subspaces of X [1]. Let ZX^DX x X be the universal subspace so that for each d e Dx, the corresponding subspace of X is given by Zx>d\=Zx n ({d} x X) £={d}xX = X. Recall that a Cartier divisor on X is a complex subspace of X whose sheaf of ideals is generated locally by a single element which is not a zero divisor. Let Div X = {d e Dx; ZXjd is a Cartier divisor on X}. Then Div X is Zariski open in Dx, and in fact is a union of connected components of Dx when X is nonsingular. Then the purpose of this paper is to prove the following: