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

基本信息

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

项目摘要

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.

投影型变化X的Castelnuovo-Mumford规律性重新定义方程式，一般的初始理想定义理想和X的希尔伯特功能。另一方面，从这个时期内，与IRRED的X型相比，预期的是，规律性与X的多样性相关，与X的范围相比，与X相比，X.字段，我们研究（i）X的多欧行；（ii）包含X的小度的高度曲面，特别是让e（x）是最多D-e+1度的所有程度性的交点，其中包含x，我们研究x = e（x）是否作为规律性猜测的证据。令b（x）为x外部的点，x的投影不是对其图像的依据。同样，令C（x）为x的平滑点，x的投影不是在其图像上的依据。我们有以下结果。（i-1）如果x是截面属G平的平滑，则x和一条线的相交的长度不超过d-e+1-g。（i-2）x和一条线的相交的长度不超过d-e+1的长度，如果x的投影从该行中的投影为quasi-finite。是quasi-finite。（ii-finite。（ii-1），如qii-finite。 B（X），C（X）和X的单位基因座（II-2）B（X）的尺寸不超过SING（x）puls 1的尺寸。