Durch Hilbertfunktionen definierte Unterschemata des Hilbertschemas. (Subschemes of the Hilbert scheme defined by Hilbert functions)
Let H be the Hilbert scheme of projective space \({\mathbb{P}}^ n\). It is well known that H is a disjoint union of subschemes \(H_ Q\) of finite type, each parametrizing coherent sheaves of ideals with a given Hilbert polynomial Q. In fact, this is the way H is constructed by \textit{A. Grothendieck} [``Techniques de construction et théorèmes d'existence en géométrie algébrique. IV'', Sém. Bourbaki 13 (1960/61), Exposé 221 (1969; Zbl 0236.14003). \textit{R. Hartshorne} proved that the \(H_ Q\) are connected [Publ. Math., Inst. Hautes Étud. Sci. 29, 5-48 (1966; Zbl 0171.415)]. The paper under review generalizes this connectedness result in the following way: For a given ideal \(I\subseteq {\mathcal O}_{P^ n}\), let h(I) be the Hilbert function of I. Introduce a partial order on the set of all functions \({\mathbb{N}}\to {\mathbb{N}}\) by putting \(f\leq g\) if and only if f(n)\(\leq g(n)\) for all \(n\in {\mathbb{N}}\). Then by standard semicontinuity results, h(I) is a semicontinuous function of I, i.e., if \(f:\quad {\mathbb{N}}\to {\mathbb{N}}\) is given, then the set \(H_{\geq f}=\{I\in H_ Q| \quad h(I)_{\geq f}\}\) is a closed subset of \(H_ Q\). The main theorem of the present paper is that the subschemes \(H_{\geq f}\) are connected in characteristic zero. - The proof makes use of the action of the upper triangular subgroup of \(GL(n+1)\) (and its subgroups) on \({\mathbb{P}}^ n\) and the induced action on \(H_ Q\). Several kinds of specializations are considered (arising from the group actions) and a careful study of the behaviour of the Hilbert function under such specialization is undertaken.
scientific article; zbMATH DE number 4071076
scientific article