Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T00:43:50.379Z Has data issue: false hasContentIssue false
  • English
  • Français

A remark on elementary contractions

Published online by Cambridge University Press:  24 October 2008

Qi Zhang
Qi Zhang
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Get access Rights & Permissions [Opens in a new window]

Extract

Let X be a smooth projective variety of dimension n over the field of complex numbers. We denote by Kx the canonical bundle of X. By Mori's theory, if Kx is not numerically effective (i.e. if there exists a curve on X which has negative intersection number with Kx), then there exists an extremal ray ℝ+[C] on X and an elementary contraction fR: X → Y associated with ℝ+[C].fR is called a small contraction if it is bi-rational and an isomorphism in co-dimension one.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 118 , Issue 1 , July 1995 , pp. 183 - 188
Copyright
Copyright © Cambridge Philosophical Society 1995

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]Alexeev, V. A.. Theorem about good divisors on log Fano varieties. Lect. Notes Math., 1479 (1989), 1–9.Google Scholar
[2]Andreatta, M., Ballico, E. and Wiśniewski, J. A.. Two theorems on elementary contractions. Math. Ann. 297 (1993), 191–198.Google Scholar
[3]Cutkosky, S.. Elementary contractions of Gorenstein threefolds. Math. Ann. 280 (1988), 521–525.CrossRefGoogle Scholar
[4]Campana, F. and Flenner, H.. Projective threefolds containing a smooth rational surface with ample normal bundle. J. reine angew. Math. 440 (1993), 77–98.Google Scholar
[5]Fujita, T.. Remarks on quasi-polarized varieties. Nagoya Math. J. 115 (1989), 105–123.CrossRefGoogle Scholar
[6]Hartshorne, R.. Algebraic Geometry (Springer-Verlag 1977).CrossRefGoogle Scholar
[7]Kawamata, Y.. Small contractions of four dimensional algebraic manifolds. Math. Ann. 284 (1989), 595–600.Google Scholar
[8]Kawamata, Y.. Crepant blowing ups of three dimensional canonical singularities, and applications to degenerations of surfaces. Ann. of Math. 127 (1988), 93–163.CrossRefGoogle Scholar
[9]Kawamata, Y., Matsuda, K. and Matsuki, K.. Introduction to the minimal model problem. Advanced Studies in Pure Mathematics 10, Algebraic Geometry, Sendai (1985).Google Scholar
[10]Maeda, H.. Ramification divisors for branched coverings of ℙn. Math. Ann. 288 (1990), 195–199.CrossRefGoogle Scholar
[11]Mobi, S.. Threefolds whose canonical bundles are not numerically effective. Ann. of Math. 116 (1982), 133–176.Google Scholar
[12]Mobi, S.. Flip theorem and the existence of minimal models for 3-folds. J. Amer. Math. Soc. 1 (1988), 117–253.Google Scholar
[13]Reid, M.. Minimal models of canonical 3-folds. Advanced Studies in Pure Mathematics 1, Algebraic Geometry, Amsterdam (1983), 131–180.Google Scholar
[14]Wiśniewski, J. A.. Length of extremal rays and generalized adjunction. Math. Z. 200 (1989), 409–427.CrossRefGoogle Scholar
[15]Wiśniewski, J. A.. On contraction of extremal rays of Fano manifolds. J. reine angew. Math. 417 (1991), 141–157.Google Scholar
[16]Wilson, P. M. H.. Towards birational classification of algebraic varieties. Bull. London Math. Soc. 19 (1987), 1–18.Google Scholar
[17]Zhang, Q.. Extremal rays on higher dimensional projective varieties. Math. Ann. 291 (1991), 497–504.CrossRefGoogle Scholar