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.

我的研究一直致力于研究投影品品种的坐标环的最低免费分辨率。特别是，Castelnuovo-Mumford的规律性是该研究项目的主要主题，我研究了Castelnuovo-Mumford的规律性的上限，从其他不变的品种中。 Castelnuovo-Mumford的规律性是最大的整数M，因此在Murnford的意义上，该品种的理想捆扎是M的，并且可以调节品种定义的定义方程式的程度，并且定义理想的共同点。 Le Tuan Hoa和我通过使用所谓的Castelnuovo结合了规律性的上限给出了一个猜想。同样，我们猜想了极端情况仅在除数的情况下发生在理性正常滚动的情况下，其程度足够大。对于曲线情况，我通过表明极端曲线的通用超平面截面在有理正常曲线中获得了极端结合。经典的caltelnuovo方法在阳性特征情况下对证明起着重要作用。此外，我已经将所有极端品种归类为在理性正常卷轴上的分隔线中的界限。这些事实应该猜测猜想。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

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

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)

A.Noma: "Castelnuovo-Mumford regularity for nonhyperelliptic curves"Archiv der Mathematik. (掲載予定).

A.Noma：“非超椭圆曲线的 Castelnuovo-Mumford 正则性”Archiv der Mathematik（待出版）。