The embedding structure, defining ideals and the projective m-normality of projective varieties

2016 Fiscal Year Final Research Report

• Project/Area Number: 26400041
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Yokohama National University
• Principal Investigator: NOMA ATSUSHI 横浜国立大学, 大学院環境情報研究院, 教授 (90262401)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Keywords: 射影多様体 / 射影埋め込み / 線形射影 / 定義方程式 / 斉次イデアル / m-正規性 / カステルヌーボマンフォード正則数

Outline of Final Research Achievements

We studied the relation between the embedding structute of projective varieties and their defining equations. For a projective variety X, a point of the projective space is called a nonbirational center of X if the linear projection from it induces a nonbirational map onto its image. By B(X) and C(X), we denote the set of nonbirational centers outside of X and inside of X respectively. We call B(X) and C(X) Segre locus and inner Segre locus respectively. In this study, first we give a characterization of a projective variety X with nonempty B(X) and C(X). As one of applications of this result, we have improved the result on Castelnuovo-Mumford regularity for smooth X of codimension e and degree d by showing that the regularity is at most e(d-e)+1. Second we show that the Castelnuovo-Mumford regularity of X with 1-dimensional C(X) is at most d-e+1. Moreover, as an application of these studies, we have a result of maximal minors of a matrix with homogeneous linear forms as entries.

Free Research Field

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