Hostname: page-component-5c6d5d7d68-xq9c7 Total loading time: 0 Render date: 2024-08-16T08:12:47.266Z Has data issue: false hasContentIssue false
  • English
  • Français

Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles

Published online by Cambridge University Press:  01 January 2008

ANTONIO LANTERI  and
HIDETOSHI MAEDA
ANTONIO LANTERI
Affiliation:
Dipartimento di Matematica ‘F. Enriques’, Università degli Studi di Milano, Via C. Saldini, 50, I-20133 Milano, Italy. e-mail: lanteri@mat.unimi.it
HIDETOSHI MAEDA
Affiliation:
Department of Mathematical Sciences, Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan. e-mail: maeda@variety.sci.waseda.ac.jp
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 1 , January 2008 , pp. 109 - 118
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Beltrametti, M. C. and Sommese, A. J.. The adjunction theory of complex projective varieties. de Gruyter Exp. Math. 16 (de Gruyter, 1995).CrossRefGoogle Scholar
[2]Elencwajg, G.. Les fibrés uniformes de rang 3 sur B P2(B C) sont homogènes. Math. Ann 231 (1978), 217–227.Google Scholar
[3]Gaiera, B. and Lanteri, A.. Ample vector bundles with zero loci of sectional genus two. Arch. Math. (Basel) 82 (2004), 495–506.CrossRefGoogle Scholar
[4]Hartshorne, R.. Algebraic geometry. Graduate Texts in Math. 52 (Springer, 1977).Google Scholar
[5]Ionescu, P.. Embedded projective varieties of small invariants. Algebraic geometry, Bucharest, 1982 (ed. Bâdescu, L. and Popescu, D.), Lecture Notes in Math. 1056 (Springer, 1984), pp. 142–186.Google Scholar
[6]Lanteri, A. and Maeda, H.. Ample vector bundle characterizations of projective bundles and quadric fibrations over curves. Higher dimensional complex varieties, Trento, 1994 (ed. Andreatta, M. and Peternell, T.), (de Gruyter, 1996), pp. 247–259.Google Scholar
[7]Lanteri, A. and Maeda, H.. Special varieties in adjunction theory and ample vector bundles. Math. Proc. Camb. Phil. Soc 130 (2001), 61–75.CrossRefGoogle Scholar
[8]Lanteri, A. and Maeda, H.. Ample vector bundles and Bordiga surfaces. Math. Nachr 280 (2007), 302–312.CrossRefGoogle Scholar
[9]Lanteri, A. and Maeda, H.. Ample vector bundles with sections vanishing on submanifolds of sectional genus three. Algebra, geometry and their interactions (ed. A. Corso, J. Migliore and C. Polini), Contemp. Math. 448 (Amer. Math. Soc., 2007), pp. 165–182.CrossRefGoogle Scholar
[10]Lanteri, A. and Novelli, C.. Ample vector bundles with zero loci of small Δ-genera, To appear in Adv. Geom.Google Scholar
[11]Maeda, H.. Ample vector bundles of small curve genera. Arch. Math. (Basel) 70 (1998), 239–243.Google Scholar
[12]Maeda, H.. Remarks on ample vector bundles of curve genus two. Arch. Math. (Basel) 88 (2007), 419–424.CrossRefGoogle Scholar
[13]Maeda, H. and Sommese, A. J.. Very ample vector bundles of curve genus two. Arch. Math. (Basel) 79 (2002), 74–80.Google Scholar
[14]Maeda, H. and Sommese, A. J.. Notes on very ample vector bundles on 3-folds. Arch. Math. (Basel) 85 (2005), 527–537.Google Scholar
[15]Okonek, C., Schneider, M. and Spindler, H.. Vector bundles on complex projective spaces. Progr. Math 3 (Birkhäuser Boston, 1980).Google Scholar