Castelnuovo's regularity and cohomological properties of sets of points in \({\mathbb{P}}^ n\)
The aim of this paper is to prove new Castelnuovo bounds for arbitrary subschemes of \({\mathbb{P}}_ k^ n\) over an algebraically closed field k. We follow an idea initiated by \textit{E. Ballico} [Boll. Un. Mat. Ital., VI. Ser., A 5, 243-246 (1986; Zbl 0601.14041)]. In some sense the heart of the paper is the following statement: Let \(\Lambda\) be a graded k- algebra of Krull-dimension \(d+1\) with \(d\geq 1\). Let P be the irrelevant ideal of A. Let \(\ell_ 1,...,\ell_ d\) be a part of a reducing system of parameters of forms of degree 1 of A. Let c be an integer such that \(c\geq 1+\sup \{t\in {\mathbb{Z}}\) with \([H^ 1_ P(A/(\ell_ 1,...,\ell_ d)A)]_ t\neq 0\}.\) We set \(j_ c(A):= \sum^{d}_{i=0}card\{p\in {\mathbb{Z}}| \quad p\geq c-i\) and \([H^ i_ P(A)]_ p\neq 0\},\) and \(\bar A:=A/H^ 0_ P(A)\). Then we get for Castelnuovo's regularity \(reg(A)\leq c-1+j_ c(A)\) if \(j_ c(\bar A)\neq 0;\quad reg(A)\leq c+j_ c(A)\) if \(j_ c(\bar A)=0\}.\) This statement has interesting consequences. For instance, by proving bounds for the integer c we improve and extends known results.
scientific article; zbMATH DE number 4063249
scientific article