Some moduli spaces of smooth complete intersections are quasi-projective
Let \(N\geq2\), \(1\leq c\leq N-1\) and \(2\leq d_1\leq \cdots\leq d_c\) be integers. Let \(H\) be the open subscheme of the appropriate Hilbert scheme of \({\mathbb P}^N_{\mathbb Z}\) parametrising smooth complete intersections of codimension \(c\) defined by homogeneous polynomials of degrees \(d_1,\cdots, d_c\). Except in the case \(c=1\), \(d_1=2\), the natural action of \(\text{PGL}_{N+1}\) on \(H\) is proper. A theorem of \textit{S. Keel} and \textit{S. Mori} [Ann. Math. (2) 145, No. 1, 193--213 (1997; Zbl 0881.14018)] shows the existence of a unique geometric quotient \(M\) of \(H\) by this action. This is a separated algebraic space of finite type over \(\text{Spec}({\mathbb Z})\). When \(c=1\) and \(d_1\geq3\), this was proved to be an affine scheme by Mumford and the proof extends easily to the case \(d_1=\cdots= d_c\) (Theorem 1.1(i)). The main purpose of this paper is to show that, if \(c\geq2\), \(d_1<d_2=\cdots= d_c\) and \(d_2(N-c+2)>d_1((c-1)(d_2-d_1)+1)\), then \(M\) is a quasi-projective scheme (Theorem 1.1 (ii)).NEWLINENEWLINEThe proof of Mumford depends on the observation that \(H\) is an open subset of a projective scheme whose complement (the discriminant divisor) is ample; hence \(H\) is affine and the result follows. This does not generalise to the case \(d_1<d_2=\cdots =d_c\); in fact, when \(c=2\), the discriminant divisor is never ample. However, \(H\) does admit an explicit compactification \(\overline{H}\); one can therefore apply GIT if one can find an ample divisor on \(\overline{H}\) such that all smooth complete intersections are stable. The restriction on degrees in the statement of (Theorem 1.1 (ii)) is required in order to prove the existence of this ample divisor.
scientific article; zbMATH DE number 6363152