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

DOI:
https://doi.org/10.1090/S0002-9947-04-03594-9

Published electronically:
December 9, 2004

MathSciNet review:
2146638

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Abstract | References | Similar Articles | Additional Information

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 of degree meets with a linear subspace of dimension at finite points, then as a finite scheme. A linear subspace for which the above length attains maximal possible value is called an *extremal* secant subspace and such for which is called a *next to extremal* secant subspace.

In this paper, we show that if a smooth variety of degree has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus . 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

Email:
sjkwak@math.kaist.ac.kr

DOI:
https://doi.org/10.1090/S0002-9947-04-03594-9

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