Bounds on Castelnuovo-Mumford regularity for graded modules and projective varieties
Let $S = k[x_0, \ldots, x_n]$ be the polynomial ring over a field $k$ and let $\text{Proj} (S) = \mathbb{P}^n_k$. The author studies the Castelnuovo-Mumford regularity of a finitely generated graded $S$-module $M$ and equivalently of a coherent sheaf $\mathcal{F}$ of $\mathbb{P}^n_k$ under the additional assumption that $M$ is equi-dimensional and locally Cohen-Macaulay. Let $V \subset \mathbb{P}^n_k$ be a $d$-dimensional projective scheme such that $(\star) \; \mathfrak{m}^v H^i_{\star}(\mathcal{I}_V) = 0$ for $i = 1,\ldots,\dim V$. Motivated by the work of \textit{U. Nagel} and the reviewer (see [Nagoya Math. J. 152, 153--174 (1998; Zbl 0945.13008)]) and the author and \textit{W. Vogel} [J. Algebra 185, No. 3, 626--642 (1996; Zbl 0912.13007)] the author studies bounds of the following type \[ \text{reg} V \leq \lceil (\deg V -1)/\text{codim}V\rceil + C(d,v). \] Note that $C(d,v) = 1$ for the case that $V$ is arithmetically Buchsbaum. As a main result it is shown that $C(d,v) = (v-1) \dim V +1$, whenever $(\star)$ is satisfied for any proper intersection $V_j = V \cap H_1 \cap \ldots \cap H_j$ of generic hyperplanes. As a technical tool the author studies Koszul homology and the highest non-vanishing degree of $H^i_{\star}(\mathcal{I}_V)$.
scientific article; zbMATH DE number 7029748
scientific article