On zero-dimensional subschemes of a complete intersections
The main aim of the paper is to consider the Hilbert function of a zero- dimensional subscheme \(X \subset \mathbb{P}^n\), \(n \geq 3\). The authors extend classical results of a finite set of points \(X \subset \mathbb{P}^2\) to the case of \(n \geq 3\), where the Hilbert-Burch structure theorem, crucial for \(n = 2\), is no longer available. One of the main results is the following: Let \(X\) be nondegenerate, and let \(a_1 \leq \cdots \leq a_r\) be the degrees of a homogeneous minimal basis of the defining ideal of \(X\). Suppose that \(X\) lies on a complete intersection of \(n - 1\) hypersurfaces of degree \(a_1 \leq \cdots \leq a_{n - 1}\) such that \(a_n \geq a_1 + \cdots + a_{n - 1} - n\). Then the \(h\)-vector of \(X\) is unimodal. As a consequence the authors obtain upper bounds for the minimal numbers of generators of homogeneous ideals, extending the classical results of \textit{P. Dubreil} [see Bull. Soc. Math. Fr. 61, 258-283 (1933; Zbl 0008.12903)] to zero dimensional schemes in \(\mathbb{P}^n\). The techniques for the proofs grow out of subtle and interesting investigations about Hilbert functions.
Hilbert function of a zero-dimensional subscheme
scientific article; zbMATH DE number 784108
scientific article