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Transactions of the American Mathematical Society

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

Author: Sijong Kwak
Journal: Trans. Amer. Math. Soc. 357 (2005), 3553-3566
MSC (2000): Primary 14M07, 14N05, 14J30
Published electronically: December 9, 2004
MathSciNet review: 2146638
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Abstract: 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 $X^{n}\subset \mathbb{P}^{n+e}$ of degree $d$meets with a linear subspace $L$ of dimension $\beta $ at finite points, then $\operatorname{length} {(X\cap L)}\le 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 $\operatorname{length}{(X\cap L)}= d-e+\beta -1$ is called a next to extremal secant subspace.

In this paper, we show that if a smooth variety $X$ of degree $d\ge 6$ has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus $g\le 1$. 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.

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Additional Information

Sijong Kwak
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Gusung-dong, Yusung-gu, Taejeon, Korea

Received by editor(s): July 3, 2003
Received by editor(s) in revised form: December 3, 2003
Published electronically: December 9, 2004
Additional Notes: This work was supported by grant No. R02-2001-000-00004 from the Korea Science and Engineering Foundation (KOSEF)
Article copyright: © Copyright 2004 American Mathematical Society