Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces

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We intend to give a classification of smooth nondegenerate projective varieties admitting extremal or next to extremal curvilinear secant subspaces. Gruson, Lazarsfeld and Peskine classified all projective integral curves with extremal secant lines. On the other hand, if a locally Cohen-Macaulay variety X-n subset of Pn+e of degree d meets with a linear subspace L of dimension beta at finite points, then length ( X boolean AND L) <= d - e + beta as a finite scheme. A linear subspace L for which the above length attains maximal possible value is called an extremal secant subspace and such L for which length ( X boolean AND L) = d - e+beta is called a next to extremal secant subspace. In this paper, we show that if a smooth variety X of degree d = 6 has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus g <= 1. This generalizes the results of Gruson, Lazarsfeld and Peskine ( 1983) for curves and the work of M-A. Bertin ( 2002) who classified smooth higher dimensional varieties with extremal secant lines. This is also motivated and closely related to establishing an upper bound for the Castelnuovo-Mumford regularity and giving a classification of the varieties on the boundary.
Publisher
AMER MATHEMATICAL SOC
Issue Date
2005
Language
English
Article Type
Article
Keywords

CASTELNUOVO; REGULARITY; CURVES; SPACE

Citation

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v.357, no.9, pp.3553 - 3566

ISSN
0002-9947
DOI
10.1090/S0002-9947-04-03594-9
URI
http://hdl.handle.net/10203/90625
Appears in Collection
MA-Journal Papers(저널논문)
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