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Bounds on cohomology and Castelnuovo-Mumford regularity. (English) Zbl 0912.13007
Let $$X\subseteq \mathbb{P}^N_K$$ be a projective scheme over an algebraically closed field $$K$$. We denote by $${\mathcal I}_X$$ the ideal sheaf of $$X$$. Then $$X$$ is said to be $$m$$-regular if $$H^i (\mathbb{P}^N_K, {\mathcal I}_X(m-i)) =0$$ for all $$i\geq 1$$. The Castelnuovo-Mumford regularity $$\text{reg} (X)$$ of $$X \subseteq \mathbb{P}^N_K$$, first introduced by Mumford by generalizing ideas of Castelnuovo, is the least such integer $$m$$. The interest in this concept stems partly from the well-known fact that $$X$$ is $$m$$-regular if and only if for every $$p\geq 0$$ the minimal generators of the $$p$$-th syzygy module of the defining ideal $$I$$ of $$X \subseteq \mathbb{P}^N_K$$ occur in degree $$\leq m+p$$. There are good bounds in some cases if $$X$$ is assumed to be smooth.
Our interest is to consider the case where $$X$$ is locally Cohen-Macaulay and equidimensional. Under this assumption there is a nonnegative integer $$k$$ such that $(X_0, \dots, X_N)^k\oplus\Bigl[ {\underset l\in\mathbb{Z} \bigoplus} H^i\bigl( \mathbb{P}_K^N, {\mathcal I}_X(l) \bigr) \Bigr]= 0\quad \text{for }1\leq i\leq \dim X,$ where $$\mathbb{P}^N_K =\text{Proj} K[X_0, \dots, X_N]$$. In this case $$X \subseteq \mathbb{P}_K^N$$ is called a $$k$$-Buchsbaum scheme. A refined version is a $$(k,r)$$-Buchsbaum scheme: Let $$k$$ and $$r$$ be integers with $$k\geq 0$$ and $$1\leq r\leq\dim X$$. Then we call $$X \subseteq \mathbb{P}^N_K$$ a $$(k,r)$$-Buchsbaum scheme if, for all $$j=0, \dots, r-1$$, $$X \cap V$$ is a $$k$$-Buchsbaum scheme for every $$(N-j)$$-dimensional complete intersection $$V$$ in $$\mathbb{P}^N_K$$ with $$\dim (X\cap V) =\dim(X) -j$$.
In recent years upper bounds on the Castelnuovo-Mumford regularity of such a variety $$X\subseteq\mathbb{P}^N_K$$ have been given by several authors in terms of $$\dim(X)$$, $$\deg(X)$$, $$k$$, and $$r$$. These bounds are stated as follows; $\text{reg}(X) \leq\left \lceil {\deg(X)- 1\over \text{codim} (X)}\right \rceil + C(k,r, d),$ where $$d=\dim X$$, $$C(k,r,d)$$ is a constant depending on $$k, r$$, and $$d$$, and $$\lceil n\rceil$$ is the smallest integer $$l\geq n$$ for a rational number $$n$$. In case $$X$$ is arithmetically Cohen-Macaulay, that is, $$k=0$$, it is well known that $$C(k,r,d)\leq 1$$. We assume $$k\geq 1$$. In case $$r=1$$ it was shown that $$C(k,1,d) \leq {d+1 \choose 2} k-d+1$$ and improved to $$C(k,1,d) \leq(2d-1) k-d+1$$. Further, a better bound $$C(k,1,d) \leq dk$$ was obtained U. Nagel and P. Schenzel [“Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity”, MPI/94-31, preprint]. The general case $$r\geq 1$$ was first studied in Le Tuan Hoa, R. M. Mirò-Roig, M. Rosa and W. Vogel [Hiroshima Math. J. 24, No. 2, 299-316 (1994; Zbl 0822.14023)] which was improved by Lê Tuân Hoa and W. Vogel [J. Algebra 163, No. 2, 348-365 (1994; Zbl 0816.14023)] by showing that $$C(k,r,d)\leq(r-1)k+{d+2-r\choose 2}k-d+1$$.
The purpose of this paper is to give bounds on $$\text{reg}(X)$$ in terms of $$\dim(X)$$, $$\deg(X)$$, $$k$$, and $$r$$, which improve some of the previous results. In the general case we show that $$C(k,r,d) \leq dk-r+1$$. Moreover, in case $$k=1$$, we show that $$C(1,r,d) \leq \left \lceil {d\over r} \right \rceil$$.
Our method here is to use a spectral sequence theory for graded modules developed by C. Miyazaki [Tokyo J. Math. 12, No. 1, 1-20 (1989; Zbl 0696.13016); J. Pure Appl. Algebra 85, No. 2, 143-161 (1993; Zbl 0773.13001); and in “Commutative Algebra 1992 ICTP, Trieste, 164-176 (1994)] in order to get bounds on the local cohomology and the Castelnuovo-Mumford regularity.

##### MSC:
 13D45 Local cohomology and commutative rings 13C14 Cohen-Macaulay modules 14F17 Vanishing theorems in algebraic geometry 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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