A study of three codimensional graded Buchsbaum domains

1. 本文编制了一个计算机程序，使我们能够利用多项式环上三维Cohen-Macaulay齐次理想的一般初始理想的基本序列，找到最小生成系统的候选项。它是有用的，因为定义一个分级Buchsbaum环的三个共维齐次理想的基本序列的前三个部分与定义一个分级Cohen-Macaulay环的三个共维齐次理想的基本序列重合。本文试图在四维射影空间的非奇异超曲面上，利用通常的Bourbaki序列与取地面场最小自由分辨率得到的最大Buchsbaum梯度模相关联，构造一个整数算术Buchsbaum曲线。根据我们过去的结果，我们知道了算术布赫斯鲍姆曲线的所有可能的基本序列，但是我们还不能用这种方法来检验哪些序列对应于积分曲线。困难似乎来自于对R/(f).3上的最大Cohen-Macaulay模缺乏完整的知识。为了重新考虑两个余维的情况，我们过去在三维射影空间中得到的关于积分曲线基本序列的许多非正式结果，现在都写在了一篇论文中。进一步将上述计算机程序应用于本例，我们发现Cook提出的条件与本文所描述的一组条件共同构成了三维射影空间中积分曲线基本序列满足的一组较好的必要条件。

1. A computer program has been made that enables us to find the candidates for the system of minimal generators of the generic initial ideal of a three codimensional Cohen-Macaulay homogeneous ideal in a polynomial ring, with the use of its basic sequence. It is useful because the first three parts of the basic sequence of a three codimensional homogeneous ideal defining a graded Buchsbaum ring coincides with the basic sequence of a three codimensional homogeneous ideal defining a graded Cohen-Macaulay ring.2. We have been trying to construct an integral arithmetically Buchsbaum curve on a nonsingular hypersurface in the four dimensional projective space, with the help of the usual Bourbaki sequence associated with the maximal Buchsbaum graded module obtained by taking the minimal free resolution of the ground field. By our past results, we know all possible basic sequences of arithmetically Buchsbaum curves, but we have not yet been able to check by this method which sequences correspond to integral curves. The difficulty seems to come from the lack of complete knowledge of maximal Cohen-Macaulay modules on R/(f).3. In order to reconsider the two codimensional case, a lot of our informal results obtained in the past on the basic sequences of integral curves in the three dimensional projective space, have now been written in a paper. Further, applying the above mentioned computer program to this case, we have found that the condition proposed by Cook and the set of conditions described in this paper together make a better set of necessary conditions for the basic sequence of an integral curve in the three dimensional projective space to satisfy.