Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces
Author:
Sijong Kwak
Journal:
Trans. Amer. Math. Soc. 357 (2005), 35533566
MSC (2000):
Primary 14M07, 14N05, 14J30
Published electronically:
December 9, 2004
MathSciNet review:
2146638
Fulltext PDF Free Access
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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 CohenMacaulay 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 MA. 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 CastelnuovoMumford 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, 3731 Gusungdong, Yusunggu, Taejeon, Korea
Email:
sjkwak@math.kaist.ac.kr
DOI:
http://dx.doi.org/10.1090/S0002994704035949
PII:
S 00029947(04)035949
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. R02200100000004 from the Korea Science and Engineering Foundation (KOSEF)
Article copyright:
© Copyright 2004 American Mathematical Society
