Multisecant subspaces to smooth projective varieties in arbitrary characteristic

Author:
Atsushi Noma

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3985-3990

MSC (2000):
Primary 14N05, 14H45

Published electronically:
July 1, 2009

MathSciNet review:
2538558

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a projective variety of dimension , degree , and codimension , not contained in any hyperplane, defined over an algebraically closed field of arbitrary characteristic. We show that if a -dimensional linear subspace meets at the smooth locus such that is finite and locally lies on a smooth curve, then the length does not exceed for the sectional genus of .

**1.**Allen Altman and Steven Kleiman,*Introduction to Grothendieck duality theory*, Lecture Notes in Mathematics, Vol. 146, Springer-Verlag, Berlin-New York, 1970. MR**0274461****2.**Marie-Amélie Bertin,*On the regularity of varieties having an extremal secant line*, J. Reine Angew. Math.**545**(2002), 167–181. MR**1896101**, 10.1515/crll.2002.032**3.**David Eisenbud and Shiro Goto,*Linear free resolutions and minimal multiplicity*, J. Algebra**88**(1984), no. 1, 89–133. MR**741934**, 10.1016/0021-8693(84)90092-9**4.**H. Flenner, L. O’Carroll, and W. Vogel,*Joins and intersections*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR**1724388****5.**T. Fujita,*Defining equations for certain types of polarized varieties*, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 165–173. MR**0437533****6.**Takao Fujita,*Classification theories of polarized varieties*, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR**1162108****7.**L. Gruson, R. Lazarsfeld, and C. Peskine,*On a theorem of Castelnuovo, and the equations defining space curves*, Invent. Math.**72**(1983), no. 3, 491–506. MR**704401**, 10.1007/BF01398398**8.**Joe Harris,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR**1182558****9.**Robin Hartshorne,*Ample subvarieties of algebraic varieties*, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. MR**0282977****10.**Sijong Kwak,*Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces*, Trans. Amer. Math. Soc.**357**(2005), no. 9, 3553–3566. MR**2146638**, 10.1090/S0002-9947-04-03594-9**11.**Atsushi Noma,*A bound on the Castelnuovo-Mumford regularity for curves*, Math. Ann.**322**(2002), no. 1, 69–74. MR**1883389**, 10.1007/s002080100265**12.**Atsushi Noma,*Castelnuovo-Mumford regularity for nonhyperelliptic curves*, Arch. Math. (Basel)**83**(2004), no. 1, 23–26. MR**2079822**, 10.1007/s00013-003-4888-5**13.**Atsushi Noma,*Multisecant lines to projective varieties*, Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, pp. 349–359. MR**2202263****14.**A. Noma,*Multisecant lines to smooth Del Pezzo varieties*, preprint, 2007.**15.**Oscar Zariski,*Introduction to the problem of minimal models in the theory of algebraic surfaces*, Publications of the Mathematical Society of Japan, no. 4, The Mathematical Society of Japan, Tokyo, 1958. MR**0097403**

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Additional Information

**Atsushi Noma**

Affiliation:
Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, Yokohama 240-8501, Japan

Email:
noma@edhs.ynu.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09977-8

Keywords:
Secant line,
secant space,
sectional genus

Received by editor(s):
June 1, 2007

Received by editor(s) in revised form:
March 20, 2009

Published electronically:
July 1, 2009

Additional Notes:
This work was partially supported by the Japan Society for the Promotion of Science.

Communicated by:
Ted Chinburg

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.