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

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05197
• 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): 2017-04-01 – 2022-03-31
• Keywords: 射影多様体 / 射影埋め込み / 線形射影 / 定義イデアル / 二重点因子

Outline of Final Research Achievements

We studied the relation between the embedding structure of projective varieties and their defining ideal. For a projective variety, its double point divisor is the nonisomorphic locus of the variety by the inner projection from the linear subspace spanned by its general (e-1)-points to its image. On the other hand, a nonbirational center of a projective variety is a point from which the variety is projected nonisomorphically. The locus of nonbirational centers off the variety (resp. on its smooth locus) is called outer (resp. Inner) Segre locus of the variety. We get the following two results. The first result is to show that the linear subsystem consisting of double point divisors of a projective variety has the base points in the singular locus or the inner Segre locus of the variety. The second result is to give upper bounds of the number of irreducible components of the Segre locus of a projective variety by its degree, dimension and codimension.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究で得られた結果，射影多様体の定義方程式を線形射影によって与える方法，セグレローカスの構造，２重点因子の豊富性は，射影代数幾何の観点から興味深いのみならず，今後の応用も期待でき，さらには解決の見通しの立っていないregularity予想の状況証拠や解決への糸口としても意義があると考えられる．これらの研究は，計算代数や計算代数幾何などへの応用が今後期待される．