Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T05:04:53.606Z Has data issue: false hasContentIssue false
  • English
  • Français

On the classification of 3-dimensional SL2(ℂ)-varieties

Published online by Cambridge University Press:  22 January 2016

Stefan Kebekus
Stefan Kebekus*
Affiliation:
Mathematisches Institut der Universität Bayreuth, 95440 Bayreuth, Germany, FAX: +49 (0)921/55-2785 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the present work we describe 3-dimensional complex SL2-varieties where the generic SL2-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional.

This is an ingredient of the classification [Keb99] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 157 , 2000 , pp. 129 - 147
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2000

References

[Arz98] Arzhantsev, I.V., On SL2-actions of complexity one, Izwestiya RAN, 61(4) (1998), 685–698.Google Scholar
[Băd82] Bădescu, L., On ample divisors, Nagoya Math. J., 86 (1982), 155–171.Google Scholar
[Băd84] Bădescu, L., Hyperplane Sections and Deformations, In Bădescu, L. and Popescu, D., editors, Algebraic Geometry Bucharest 1982, Lecture Notes in Mathematics, 1056, Springer, 1984.Google Scholar
[Eis95] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer, 1995.Google Scholar
[Ful93] Fulton, W., Introduction to Toric Varieties, Annals of Mathematics Studies, 131, Princeton University Press, 1993.Google Scholar
[Gro62] Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA2), Advanced studies in pure mathematics, 2, North-Holland Publ. Co., Amsterdam, 1962.Google Scholar
[HO80] Huckleberry, A. and Oeljeklaus, E., Classification Theorems for Almost Homogeneous Spaces, Institut Elie Cartan, 1980.Google Scholar
[Huc86] Huckleberry, A., The classification of homogeneous surfaces, Expo. Math., 4 (1986).Google Scholar
[Keb99] Kebekus, S., Relatively minimal quasihomogeneous projective 3-Folds, to appear (1998).Google Scholar
[Kra85] Kraft, H., Geometrische Methoden in der Invariantentheorie, Aspekte der Mathematik, D1, Vieweg, 2, durchges. Aufl., 1985.Google Scholar
[LV83] Luna, D. and Vust, T., Plongements d’espaces homogènes, Comment. Math. Helv., 58 (1983), 186–245.Google Scholar
[Mor82] Mori, S., Threefolds whose canonical bundles are not numerically effective, Ann. of Math., 116 (1982), 133–176.Google Scholar
[Mor85] Mori, S., On 3-dimensional terminal singularities, Nagoya Math. J., 98 (1985), 43–66.CrossRefGoogle Scholar
[MU83] Mukai, S. and Umemura, H., Minimal rational threefolds, In M. Raynaud and T. Shioda, editors, Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math., 1016 (1983), Springer, 490–518.Google Scholar
[Pin74] Pinkham, H., Deformations of cone with negative grading, Journal of Algebra, 30 (1974), 92–102.CrossRefGoogle Scholar
[Rei80] Reid, M., Canonical 3-folds, In Arnaud Beauville, editor, Algebraic Geometry Angers 1979, Alphen aan den Rijn, Sijthoff & Noordhoff, 1980.Google Scholar
[Rei87] Reid, M., Young person’s guide to canonical singularities, Proceedings of Symposia in Pure Mathematics, 46 (1987), 345–414.Google Scholar