Multisecant subspaces to smooth projective varieties in arbitrary characteristic
Author:
Atsushi Noma
Journal:
Proc. Amer. Math. Soc. 137 (2009), 39853990
MSC (2000):
Primary 14N05, 14H45
Published electronically:
July 1, 2009
MathSciNet review:
2538558
Fulltext PDF
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, SpringerVerlag, BerlinNew York, 1970. MR 0274461
(43 #224)
 2.
MarieAmélie
Bertin, On the regularity of varieties having an extremal secant
line, J. Reine Angew. Math. 545 (2002),
167–181. MR 1896101
(2003h:14078), http://dx.doi.org/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
(85f:13023), http://dx.doi.org/10.1016/00218693(84)900929
 4.
H.
Flenner, L.
O’Carroll, and W.
Vogel, Joins and intersections, Springer Monographs in
Mathematics, SpringerVerlag, Berlin, 1999. MR 1724388
(2001b:14010)
 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
(55 #10457)
 6.
Takao
Fujita, Classification theories of polarized varieties, London
Mathematical Society Lecture Note Series, vol. 155, Cambridge
University Press, Cambridge, 1990. MR 1162108
(93e:14009)
 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 (85g:14033), http://dx.doi.org/10.1007/BF01398398
 8.
Joe
Harris, Algebraic geometry, Graduate Texts in Mathematics,
vol. 133, SpringerVerlag, New York, 1992. A first course. MR 1182558
(93j:14001)
 9.
Robin
Hartshorne, Ample subvarieties of algebraic varieties, Lecture
Notes in Mathematics, Vol. 156, SpringerVerlag, BerlinNew York, 1970.
Notes written in collaboration with C. Musili. MR 0282977
(44 #211)
 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
(2006e:14072), http://dx.doi.org/10.1090/S0002994704035949
 11.
Atsushi
Noma, A bound on the CastelnuovoMumford regularity for
curves, Math. Ann. 322 (2002), no. 1,
69–74. MR
1883389 (2002k:14046), http://dx.doi.org/10.1007/s002080100265
 12.
Atsushi
Noma, CastelnuovoMumford regularity for nonhyperelliptic
curves, Arch. Math. (Basel) 83 (2004), no. 1,
23–26. MR
2079822 (2005c:14039), http://dx.doi.org/10.1007/s0001300348885
 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
(2006k:14099)
 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
(20 #3872)
 1.
 A. Altman and S. Kleiman, Introduction to Grothendieck duality theory, Lecture Notes in Math., 146, SpringerVerlag, 1970. MR 0274461 (43:224)
 2.
 M. A. Bertin, On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167181. MR 1896101 (2003h:14078)
 3.
 D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89133. MR 741934 (85f:13023)
 4.
 H. Flenner, L. O'Carroll, and W. Vogel, Joins and intersections, Springer Monographs in Mathematics, SpringerVerlag, 1999. MR 1724388 (2001b:14010)
 5.
 T. Fujita, Defining equations for certain types of polarized varieties, Complex analysis and algebraic geometry, Cambridge University Press, 1977, pp. 165173. MR 0437533 (55:10457)
 6.
 T. Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, 155, Cambridge University Press, 1990. MR 1162108 (93e:14009)
 7.
 L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983), 491506. MR 704401 (85g:14033)
 8.
 J. Harris, Algebraic geometry, Graduate Texts in Mathematics, 133, SpringerVerlag, 1992. MR 1182558 (93j:14001)
 9.
 R. Hartshone, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, 156, SpringerVerlag, 1970. MR 0282977 (44:211)
 10.
 S. Kwak, Smooth projective varieties with extremal or next to extremal curvilinear secant subspaces, Trans. Amer. Math. Soc. 357 (2005), 35533566. MR 2146638 (2006e:14072)
 11.
 A. Noma, A bound on the CastelnuovoMumford regularity for curves, Math. Ann. 322 (2002), 6974. MR 1883389 (2002k:14046)
 12.
 A. Noma, CastelnuovoMumford regularity of nonhyperelliptic curves, Arch. Math. (Basel) 83, no. 1 (2004), 2326. MR 2079822 (2005c:14039)
 13.
 A. Noma, Multisecant lines to projective varieties, Projective varieties with unexpected properties, Walter de Gruyter, 2005, pp. 349359. MR 2202263 (2006k:14099)
 14.
 A. Noma, Multisecant lines to smooth Del Pezzo varieties, preprint, 2007.
 15.
 O. Zariski, Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. of the Math. Soc. of Japan, no. 4, The Mathematical Society of Japan, Tokyo, 1958. MR 0097403 (20:3872)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
14N05,
14H45
Retrieve articles in all journals
with MSC (2000):
14N05,
14H45
Additional Information
Atsushi Noma
Affiliation:
Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, Yokohama 2408501, Japan
Email:
noma@edhs.ynu.ac.jp
DOI:
http://dx.doi.org/10.1090/S0002993909099778
PII:
S 00029939(09)099778
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.
