Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

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 $X^{n}\subset \mathbb{P}^{n+e}$ of degree $d$meets with a linear subspace $L$ of dimension $\beta $ at finite points, then $\operatorname{length} {(X\cap L)}\le d-e+\beta $ as a finite scheme. A linear subspace $L$ for which the above length attains maximal possible value is called an extremal secant subspace and such $L$ for which $\operatorname{length}{(X\cap L)}= d-e+\beta -1$ is called a next to extremal secant subspace.

In this paper, we show that if a smooth variety $X$ of degree $d\ge 6$ has extremal or next to extremal curvilinear secant subspaces, then it is either Del Pezzo or a scroll over a curve of genus $g\le 1$. 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.


References [Enhancements On Off] (What's this?)

  • [AB] A. Alzati, G. M. Besana, On the $k$-regularity of some projective manifolds, Collect. Math. 49 (2-3) (1998), 149-171. MR 2000a:14005
  • [ABB] A. Alzati, M. Bertolini and G. M. Besana, Projective normality of varieties of small degree, Comm. Algebra (1997), 3761-3771. MR 98j:14053
  • [Be] M.-A. Bertin, On the regularity of varieties having an extremal secant line, J. Reine Angew. Math. 545 (2002), 167-181. MR 2003h:14078
  • [Bu] D. C. Butler, Normal generation of vector bundles over a curve, J. Diff. Geom. 39 (1) (1994), 1-34. MR 94k:14024
  • [D] J. D'Almeida, Courbes de l'espace projectif: Series lineaires incompletes et multisecantes, J. Reine Angew. Math. 370 (1986), 30-51. MR 87k:14034
  • [E] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics 150, Springer-Verlag, Heidelberg, 1995. MR 97a:13001
  • [EG] D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89-133. 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. 1-13. MR 89f:14042
  • [ES] Ph. Ellia and G. Sacchiero, Smooth surfaces in $\mathbb{P}^{4}$ ruled in conics, Algebraic Geometry (Catania, 1993/Barcelona, 1994), Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, 1998, pp. 49-62.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), 491-506. MR 85g:14033
  • [GP] F. Gallego and B. Purnaprajna, Normal Presentation on Elliptic Ruled Surfaces, J. Algebra 186 (1996), 597-625. MR 98c:14030
  • [Ha] J. Harris, Algebraic Geometry, GTM 133, Springer-Verlag, 1992.MR 93j:14001
  • [Io] P. Ionescu, Embedded projective varieties of small invariants, Algebraic Geometry (Bucharest, 1982), Lecture Notes in Math., vol. 1056, Springer-Verlag, Berlin-Heidelberg-New York, 1984. MR 85m:14024
  • [K1] S. Kwak, Castelnuovo regularity of smooth projective varieties of dimension 3 and 4, J. Algebraic Geometry 7 (1998), 195-206. MR 2000d:14043
  • [K2] S. Kwak, Castelnuovo-Mumford regularity for smooth threefolds in $\mathbb{P}^{5}$ 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 Castelnuovo-Mumford regularity for curves, Math. Ann. 322 (2002), 69-74. MR 2002k:14046
  • [Oh] M. Ohno, On odd-dimensional projective manifolds with smallest secant varieties, Math. Z. 226 (3) (1997), 483-498. MR 99a:14059
  • [SV] A. J. Sommese and A. Van de Ven, On the adjunction mapping, Math. Ann. 278 (1987), 593-603. 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

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M07, 14N05, 14J30

Retrieve articles in all journals with MSC (2000): 14M07, 14N05, 14J30


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

American Mathematical Society