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Syzygies for the defining ideal of projective varieties

Syzygies 定义射影簇的理想

基本信息

批准号：
11640052
负责人：
MIYAZAKI Chikashi
金额：
$ 1.66万
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：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640052/
关键词：
Castelnuovo free resolition syzygy algebraic curve

项目摘要

My research has been devoted to the study of free resolutions for defining ideals of projective varieties, especially to that of the Castelnuovo-Mumford regularities. The regularity is a basic invatiant which describes the minimal free resolutions and the degrees of the defining equations of the varieties. Let X ⊂ P^N_K be a projective variety over an algebraically closed field K. Then, by using an invariant k(X) which evaluates the deficiency of the Hartshorne-Rao module of the variety, we have known an upper bound on the regularity reg(X) 【less than or equal】[(deg(X)-1)/codim(X)] + max { k(X)・dim(X), 1}. In order to classify the equality case, we consider a generic hyperplane section of the projective curve satisfying reg(X) = [(deg(X)-1)/codim(X)] + 1. In case char(k) = 0, the uniform position principle yields an information on the configuration of the zero-dimensional scheme, and the set of points as a generic hyperplane section is contained in a rational normal curve. In case char(k) > 0, the correspodence between the monodromy group of the projective curve and the configuration of the points excludes the strange curves. Thus, by dimensional induction, the sharp bounds are only appeared in the case of divisors on a Hirzeburch surfaces if X is not arithmetically Cohen-Macaulay. Further I have conjectured with Le Tuan Hoa, by introducing a new invariant k^^〜(X), reg(X) 【less than or equal】[(deg(X) -1)/codim(X)] + max.{ k^^〜(X), 1}, and the bound is obtained to be effective for the divisor on the rational normal scroll.

我的研究一直致力于研究自由决议的定义理想的投射品种，特别是对的Castelnuovo-Mumford的。正则性是描述簇的定义方程的最小自由分解和次数的基本不变量。设X ∈ P^N_K是代数闭域K上的射影簇.然后，利用一个不变量k（X）来计算簇的Hartshorne-Rao模的亏度，我们得到了正则性reg（X）[小于或等于][（deg（X）-1）/codim（X）] + max { k（X）·dim（X），1}的一个上界.为了对等式情况进行分类，我们考虑满足reg（X）= [（deg（X）-1）/codim（X）] + 1的射影曲线的一般超平面部分。在char（k）= 0的情况下，一致位置原理给出了零维格式的结构信息，并且作为一般超平面截面的点集包含在有理法向曲线中。当char（k）> 0时，射影曲线的单值群与点的构形的对应关系排除了奇异曲线。因此，通过维数归纳，只有当X不是算术Cohen-Macaulay时，在Hirzeburch曲面上的因子的情况下才出现尖锐的界。此外，我通过引入一个新的不变量k^^（X），reg（X）[小于或等于][（deg（X）-1）/codim（X）] + max，与Le Tuan Hoa进行了讨论。{ k^^<$（X），1}，并且得到了对有理数正规涡卷上的除数有效的界。