an:00554211
Zbl 0816.14023
L?? Tu??n Hoa; Vogel, Wolfgang
Castelnuovo-Mumford regularity and hyperplane sections
EN
J. Algebra 163, No. 2, 348-365 (1994).
00018158
1994
j
14N05 14C17 14F17 14M05 13D02
hyperplane sections; Cohen-Macaulay varieties; local cohomology; arithmetically quasi-Buchsbaum variety; Buchsbaum modules; Castelnuovo- Mumford regularity; syzygies
Let \(X \subset \mathbb{P}^ n\) be a projective variety and \({\mathcal I}_ X\) the ideal sheaf of \(X\) in \(\mathbb{P}^ n\). One says that \(X\) is \(k\)-regular if \(H^ i (\mathbb{P}^ n, {\mathcal I}_ X (k-i)) = 0\) for all \(i \geq 1\). The Castelnuovo-Mumford regularity \(\text{reg} (X)\) of \(X \subset \mathbb{P}^ n\) is the last such \(k\). Bounds on the regularity of \(X\) are important in connection with algorithms for computing syzygies and this accounts for some of the recent interest in this question.
There is a classical technique aimed at bounding \(\text{reg} (X)\) by studying \(H^ i (\mathbb{P}^ n, {\mathcal I}_{X \cap H})\) for a hyperplane section \(X \cap H\) of \(X\). By using this method (corollaries 1 and 2.3), in the present paper the authors give a new bound for \(\text{reg} (X)\) of arithmetically quasi-Buchsbaum varieties \(X \subset \mathbb{P}^ n\), in terms of their invariants (corollary 2). Recall that \(X\) is said to be arithmetically quasi-Buchsbaum if \(\bigoplus_{p \in \mathbb{Z}} H^ i (\mathbb{P}^ n, {\mathcal I}_ X (p))\) is annihilated by \((x_ 0, \dots, x_ n)\) for all \(1 \leq i \leq \dim X\). The authors obtain this result in a general form (proposition 3.6), as special case of their investigation of \((k,r)\)-Buchsbaum modules (see definition 1.1), from which they deduce other new bounds on the regularity, improve some old result and, in some case, provide better bounds as expected (e.g. theorems 3.4 and 3.8 and their example on p. 349).
V.Di Gennaro (Roma)