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.
 [AB]
Alberto
Alzati and Gian
Mario Besana, On the 𝑘regularity of some projective
manifolds, Collect. Math. 49 (1998), no. 23,
149–171. Dedicated to the memory of Fernando Serrano. MR 1677105
(2000a:14005)
 [ABB]
Alberto
Alzati, Marina
Bertolini, and Gian
Mario Besana, Projective normality of varieties of small
degree, Comm. Algebra 25 (1997), no. 12,
3761–3771. MR 1481563
(98j:14053), http://dx.doi.org/10.1080/00927879708826083
 [Be]
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
 [Bu]
David
C. Butler, Normal generation of vector bundles over a curve,
J. Differential Geom. 39 (1994), no. 1, 1–34.
MR
1258911 (94k:14024)
 [D]
Jean
d’Almeida, Courbes de l’espace projectif: séries
linéaires incomplètes et multisécantes, J. Reine
Angew. Math. 370 (1986), 30–51 (French). MR 852508
(87k:14034), http://dx.doi.org/10.1515/crll.1986.370.30
 [E]
David
Eisenbud, Commutative algebra, Graduate Texts in Mathematics,
vol. 150, SpringerVerlag, New York, 1995. With a view toward
algebraic geometry. MR 1322960
(97a:13001)
 [EG]
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
 [EH]
David
Eisenbud and Joe
Harris, On varieties of minimal degree (a centennial account),
Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos.
Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987,
pp. 3–13. MR 927946
(89f:14042)
 [ES]
Philippe
Ellia and Gianni
Sacchiero, Smooth surfaces of 𝑃⁴ ruled in
conics, Algebraic geometry (Catania, 1993/Barcelona, 1994) Lecture
Notes in Pure and Appl. Math., vol. 200, Dekker, New York, 1998,
pp. 49–62. MR 1651089
(99j:14038)
 [Fu]
Takao
Fujita, Classification theories of polarized varieties, London
Mathematical Society Lecture Note Series, vol. 155, Cambridge
University Press, Cambridge, 1990. MR 1162108
(93e:14009)
 [GLP]
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
 [GP]
Francisco
Javier Gallego and B.
P. Purnaprajna, Normal presentation on elliptic ruled
surfaces, J. Algebra 186 (1996), no. 2,
597–625. MR 1423277
(98c:14030), http://dx.doi.org/10.1006/jabr.1996.0388
 [Ha]
Joe
Harris, Algebraic geometry, Graduate Texts in Mathematics,
vol. 133, SpringerVerlag, New York, 1992. A first course. MR 1182558
(93j:14001)
 [Io]
Paltin
Ionescu, Embedded projective varieties of small invariants,
Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in
Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR 749942
(85m:14024), http://dx.doi.org/10.1007/BFb0071773
 [K1]
Sijong
Kwak, Castelnuovo regularity for smooth subvarieties of dimensions
3 and 4, J. Algebraic Geom. 7 (1998), no. 1,
195–206. MR 1620706
(2000d:14043)
 [K2]
SiJong
Kwak, CastelnuovoMumford regularity bound for smooth threefolds in
𝑃⁵ and extremal examples, J. Reine Angew. Math.
509 (1999), 21–34. MR 1679165
(2000e:14064), http://dx.doi.org/10.1515/crll.1999.040
 [K3]
S. Kwak, Multisecant spaces to projective varieties, Preprint.
 [KP]
S. Kwak and E. Park, Regularity and higher normality of ruled varieties over curves, Preprint.
 [Mu]
David
Mumford, Lectures on curves on an algebraic surface, With a
section by G. M. Bergman. Annals of Mathematics Studies, No. 59, Princeton
University Press, Princeton, N.J., 1966. MR 0209285
(35 #187)
 [No]
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
 [Oh]
Masahiro
Ohno, On odddimensional projective manifolds with smallest secant
varieties, Math. Z. 226 (1997), no. 3,
483–498. MR 1483544
(99a:14059), http://dx.doi.org/10.1007/PL00004352
 [SV]
Andrew
John Sommese and A.
Van de Ven, On the adjunction mapping, Math. Ann.
278 (1987), no. 14, 593–603. MR 909240
(88j:14011), http://dx.doi.org/10.1007/BF01458083
 [Z]
F.
L. Zak, Tangents and secants of algebraic varieties,
Translations of Mathematical Monographs, vol. 127, American
Mathematical Society, Providence, RI, 1993. Translated from the Russian
manuscript by the author. MR 1234494
(94i:14053)
 [AB]
 A. Alzati, G. M. Besana, On the regularity of some projective manifolds, Collect. Math. 49 (23) (1998), 149171. MR 2000a:14005
 [ABB]
 A. Alzati, M. Bertolini and G. M. Besana, Projective normality of varieties of small degree, Comm. Algebra (1997), 37613771. MR 98j:14053
 [Be]
 M.A. Bertin, On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167181. MR 2003h:14078
 [Bu]
 D. C. Butler, Normal generation of vector bundles over a curve, J. Diff. Geom. 39 (1) (1994), 134. MR 94k:14024
 [D]
 J. D'Almeida, Courbes de l'espace projectif: Series lineaires incompletes et multisecantes, J. Reine Angew. Math. 370 (1986), 3051. MR 87k:14034
 [E]
 D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, SpringerVerlag, Heidelberg, 1995. MR 97a:13001
 [EG]
 D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89133. MR 85f:13023
 [EH]
 D. Eisenbud and J. Harris, Varieties of minimal degree (a centennial account), Bowdoin conference in Algebraic Geometry (Brunswick, Maine 1985), Proc. Symp. Pure Math., vol. XXXXVI, part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 113. MR 89f:14042
 [ES]
 Ph. Ellia and G. Sacchiero, Smooth surfaces in ruled in conics, Algebraic Geometry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, 1998, pp. 4962.MR 99j:14038
 [Fu]
 T. Fujita, Classification theories of polarized varieties, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR 93e:14009
 [GLP]
 L. Gruson, R. Lazarsfeld and C. Peskine, On a theorem of Castelnuovo and the equations defining projective varieties, Inv. Math. 72 (1983), 491506. MR 85g:14033
 [GP]
 F. Gallego and B. Purnaprajna, Normal Presentation on Elliptic Ruled Surfaces, J. Algebra 186 (1996), 597625. MR 98c:14030
 [Ha]
 J. Harris, Algebraic Geometry, GTM 133, SpringerVerlag, 1992.MR 93j:14001
 [Io]
 P. Ionescu, Embedded projective varieties of small invariants, Algebraic Geometry (Bucharest, 1982), Lecture Notes in Math., vol. 1056, SpringerVerlag, BerlinHeidelbergNew York, 1984. MR 85m:14024
 [K1]
 S. Kwak, Castelnuovo regularity of smooth projective varieties of dimension 3 and 4, J. Algebraic Geometry 7 (1998), 195206. MR 2000d:14043
 [K2]
 S. Kwak, CastelnuovoMumford regularity for smooth threefolds in and extremal examples, J. Reine Angew. Math. 509 (1999).MR 2000e:14064
 [K3]
 S. Kwak, Multisecant spaces to projective varieties, Preprint.
 [KP]
 S. Kwak and E. Park, Regularity and higher normality of ruled varieties over curves, Preprint.
 [Mu]
 D. Mumford, Lectures on curves on an algebric surface, Annals of Math. Studies, vol. 59, 1966. MR 35:187
 [No]
 A. Noma, A bound on the CastelnuovoMumford regularity for curves, Math. Ann. 322 (2002), 6974. MR 2002k:14046
 [Oh]
 M. Ohno, On odddimensional projective manifolds with smallest secant varieties, Math. Z. 226 (3) (1997), 483498. MR 99a:14059
 [SV]
 A. J. Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann. 278 (1987), 593603. MR 88j:14011
 [Z]
 Fyodor Zak, Tangents and secants of algebraic varieties, Translation of Math. Monographs, vol. 127, Amer. Math. Soc., 1993. MR 94i:14053
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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
