Embedding structure of projective varieties and the initial ideal of their definig equations
| Project/Area Number |
17540017
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Yokohama National University |
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Principal Investigator |
NOMA Atsushi Yokohama National University, Faculty of Education and Human Sciences, Associate Professor (90262401)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | defining equation / projective embedding / projective varieity / Castelnuovo-Mumford regularity / hypersurface / linear projection / secant line / hypersurface / Castelnuovo-Mumford reularity / sectional genus |
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| Research Abstract |
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.
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Report
(4 results)
Research Products
(18 results)
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[Presentation] Hypersurfaces cutting out a projective variety2006
Author(s)
Atsushi, NOMA
Organizer
Workshop on Syzygies and Hilbert Functions
Place of Presentation
BanffInternational Research Station for Mathematical Innovation and Discovery at Banff Center, Banff, Canada
Year and Date
2006-10-19
Description
「研究成果報告書概要(欧文)」より
Related Report
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