Free resolutions of the coordinate rings of projective varieties

射影簇坐标环的自由分辨率

• 批准号: 14540036
• 负责人: MIYAZAKI Chikashi
• 金额: $ 2.24万
• 依托单位: University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540036/
• 关键词: Castelnuovo-Mumford regularity; syzygy; free resolution; projective curve; rational normal scroll; Castelnuovo; 代数曲線; 多項式イデアル; 自由分解; 射影束

My research has been devoted to the study of minimal free resolutions for the coordinate rings of projective varieties. In particular, the Castelnuovo-Mumford regularity is the main theme of this research project, and I have studied the upper bounds of the Castelnuovo-Mumford regularity in terms of the other invariants of give varieties. The Castelnuovo-Mumford regularity is the maximal integer m such that the ideal sheaf of the varietiy is m-regular in Murnford sense, and it regulates the degree of the defining equations of the variety and the syzygies of the defining ideal of the variety. Le Tuan Hoa and I had given a conjecture on an upper bound of the regularity by using so-called the Castelnuovo bound. Also we had conjectured the extremal case happens only in the case of a divisor on a rational normal scroll for its degree large enough. For the curve case, I have obtained an extremal bound by showing that the generic hyperplane section of an extremal curve is contained in a rational normal curve. The classical Caltelnuovo method plays an important role for the proof in positive characteristic case. Moreover, I have classified all the extremal varieties for the bound among the divisors on rational normal scrolls. These facts should guess the conjecture. I would like to extend them to the corresponding results for weighted projective space.

我的研究致力于射影簇坐标环的最小自由分辨率的研究。特别是，Castelnuovo-Mumford 正则性是本研究项目的主题，我根据给定簇的其他不变量研究了 Castelnuovo-Mumford 正则性的上限。 Castelnuovo-Mumford正则性是使品种的理想层在Murnford意义上是m正则的最大整数m，它调节品种的定义方程的次数和品种的定义理想的syzygies。 Le Tuan Hoa 和我通过使用所谓的 Castelnuovo 界，对正则性的上限做出了猜想。我们还推测，极值情况仅发生在有理法向卷轴上的除数足够大的情况下。对于曲线情况，我通过证明极值曲线的通用超平面部分包含在有理正态曲线中，获得了极值界。经典的Caltelnuovo方法对于正特征情况的证明起着重要作用。此外，我已经将所有极值簇分类为有理正卷轴上除数之间的界限。这些事实应该是猜测的猜想。我想将它们扩展到加权投影空间的相应结果。

项目成果

Verification of the Connectedness of space curve invariants for a special case

特殊情况下空间曲线不变量连通性的验证

• 发表时间: 2004
• 期刊: Communications in Algebra 32
• 作者: Mutsumi Amasaki

Maximal Buchsbaum modules over Gorenstein local rings

Gorenstein 局部环上的最大 Buchsbaum 模

• 发表时间: 2005
• 期刊: J.Algebra 287
• 作者: Juergen Herzog;Yukihide Takayama;Naoki Terai;H.Katsurada;Mutsumi Amasaki
• 通讯作者: Mutsumi Amasaki

T.Maeda: "A partial order on the symmetric groups defined by 3-cycles"Ryukyu Math.J.. 15. 19-42 (2002)

T.Maeda：“由 3 圈定义的对称群的偏序”Ryukyu Math.J.. 15. 19-42 (2002)

Bounds on Castelnuovo-Mumford regularity for divisors on rational normal scroll

有理法向滚动除数的 Castelnuovo-Mumford 正则性的界限

• 发表时间: 2005
• 期刊: Collect.Math. 56
• 作者: Chikashi Miyazaki

The varieties of subspaces stable under a nilpotent transformation

幂零变换下稳定的子空间簇

• 发表时间: 2003
• 期刊: Ryukyu Math.J. 16
• 作者: A.Noma;Takashi Maeda
• 通讯作者: Takashi Maeda