Embedding structure of projective varieties and the initial ideal of their definig equations

射影簇的嵌入结构及其定义方程的初始理想

• 批准号: 17540017
• 负责人: NOMA Atsushi
• 金额: $ 2.24万
• 依托单位: Yokohama National University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540017/
• 关键词: defining equation; projective embedding; projective varieity; Castelnuovo-Mumford regularity; hypersurface; linear projection; secant line; hypersurface; Castelnuovo-Mumford reularity; sectional genus

The Castelnuovo-Mumford regularity of a projective varierty X refrects its defining equations, generic initial ideal of defining ideal, and the Hilbert function of X. On the other hand, the regularity is expected to have a strong relation to the existence of multisecant lines to X. From this point of view, in this period, for an irreducible, projective variety X of degree d and codimension e, defined over an algebraically closed field, we study (I) multisecant lines to X; (II) hypersurfaces of small degree containing X. In (II), in particular, letting E(X) be the intersection of all hypersurfaces of degree at most d-e+1, containing X, we study if X = E(X) as an evidence of the regularity conjecture. Let B(X) be the points of outside of X, from which the projection of X is not birational onto its image. Similarly, let C(X) be the smooth points of X, from which the projection of X is not birational onto its image. We have the following results.(I-1) If X is smooth of sectional genus g, the length of the intersection of X and a line does not exceed d-e+1-g.(I-2) The length of the intersection of X and a line does not exceed d-e+1 if the projection of X from the line is quasi-finite.(II-1) As sets, X=E(X) outside of B(X), and as schemes, X=E(X) outside of B(X), C(X) and the singular locus Sing(X) of X.(II-2) The dimension of B(X) does not exceed the dimension of Sing(X) puls 1. Moreover, the dimension of C(X) does not exceed the dimension of Sing(X) plus 2.

射影射变 X 的 Castelnuovo-Mumford 正则性反映了它的定义方程、定义理想的一般初始理想以及 X 的希尔伯特函数。另一方面，该正则性预计与 X 的多重割线的存在性有很强的关系。从这个角度来看，在这个时期，对于一个不可约的、度数为 d 且余维数为 e 的射影射影 X，在代数闭上定义 领域，我们研究 (I) X 的多重割线； (II)包含X的小次数超曲面。在(II)中，特别地，令E(X)为所有包含X的次数最多为d-e+1的超曲面的交集，我们研究是否X = E(X)作为正则猜想的证据。设 B(X) 为 X 外部的点，从这些点 X 到其图像的投影不是双有理的。类似地，令 C(X) 为 X 的平滑点，由此 X 到其图像的投影不是双有理的。我们得到以下结果： (I-1) 如果 X 是截面亏格 g 光滑的，则 X 与直线的交点长度不超过 d-e+1-g。 (I-2) 如果 X 从直线的投影是拟有限的，则 X 与直线的交点长度不超过 d-e+1。 (II-1) 作为集合，X=E(X) 在 B(X) 之外，作为方案，X=E(X)外面 B(X)、C(X)和X的奇异轨迹Sing(X)。 (II-2) B(X)的维数不超过Sing(X)脉冲1的维数。此外，C(X)的维数不超过Sing(X)加2的维数。

项目成果

Hyper surfaces cutting out a projective variety

超曲面切割射影变化

• 发表时间: 2008
• 作者: 佐藤文広;小木曽岳義;Atsushi NOMA;小木曽岳義;Atsushi Noma;小木曽岳義;Atsushi Noma;Atsushi Noma;小木曽岳義;Atsushi Noma;Atsushi Noma;Atsushi Noma;Atsushi Noma;Atsushi Noma;Atsushi Noma;浅沼照雄;Atsushi Noma;浅沼照雄;Atsushi Noma;Atushi Noma;Atsushi NOMA;Atsushi Noma;Atsushi NOMA;浅沼照雄;Atsushi Noma;Atsushi NOMA
• 通讯作者: Atsushi NOMA

Hypersurfaces cutting out a projective varieity

超曲面切出射影多样性

• 发表时间: 2007
• 作者: NOMA;Atsushi
• 通讯作者: Atsushi

Very ample line bundles on regular surfaces obtained by projection

通过投影获得的规则表面上非常丰富的线束

• 发表时间: 2006
• 期刊: Journal of Algebra 306・2
• 作者: Natsuo;Saito;齋藤 夏雄;廣門 正行;齋藤 夏雄;伊藤 浩行;廣門 正行;齋藤 夏雄;Atsushi NOMA
• 通讯作者: Atsushi NOMA

Hypersurfaces cutting out a projective variety and the centers of nonbirational linear projections

超曲面切除射影变化和非双理线性投影的中心

• 发表时间: 2007
• 作者: Atsushi;NOMA
• 通讯作者: NOMA