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Projective normality of finite group quotients


Authors: S. S. Kannan, S. K. Pattanayak and Pranab Sardar
Journal: Proc. Amer. Math. Soc. 137 (2009), 863-867
MSC (2000): Primary 14Lxx
DOI: https://doi.org/10.1090/S0002-9939-08-09613-5
Published electronically: September 15, 2008
MathSciNet review: 2457424
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Abstract: In this paper, we prove that for any finite dimensional vector space $ V$ over an algebraically closed field $ k$, and for any finite subgroup $ G$ of $ GL(V)$ which is either solvable or is generated by pseudo reflections such that $ \vert G\vert$ is a unit in $ k$, the projective variety $ \mathbb{P}(V)/G$ is projectively normal with respect to the descent of $ \mathcal O(1)^{\otimes \vert G\vert}$.


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

  • 1. Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 0072877, https://doi.org/10.2307/2372597
  • 2. P. Erdös, A. Ginzburg, and A. Ziv, A theorem in additive number theory, Bull. Res. Council, Israel, 10 F (1961), 41-43.
  • 3. Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • 4. Richard Kane, Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 5, Springer-Verlag, New York, 2001. MR 1838580
  • 5. Hanspeter Kraft, Peter Slodowy, and Tonny A. Springer (eds.), Algebraische Transformationsgruppen und Invariantentheorie, DMV Seminar, vol. 13, Birkhäuser Verlag, Basel, 1989 (German). MR 1044582
  • 6. D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
  • 7. P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51, Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi, 1978. MR 546290
  • 8. Emmy Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann. 77 (1915), no. 1, 89–92 (German). MR 1511848, https://doi.org/10.1007/BF01456821
  • 9. E. Noether, Der Endlichkeitssatz der Invarianten endlich linearer Gruppen der Charakteristik $ p$, Nachr. Akad. Wiss. Gottingen (1926), 28-35.
  • 10. J-P. Serre, Groupes finis d'automorphismes d'anneaux locaux réguliers, Colloq. d'Alg. École Norm. de Jeunes Filles, Paris (1967), 1-11.
  • 11. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304. MR 0059914

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Additional Information

S. S. Kannan
Affiliation: Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur Post Office, Siruseri, Tamilnadu, 603103, India
Email: kannan@cmi.ac.in

S. K. Pattanayak
Affiliation: Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur Post Office, Siruseri, Tamilnadu, 603103, India
Email: santosh@cmi.ac.in

Pranab Sardar
Affiliation: Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur Post Office, Siruseri, Tamilnadu, 603103, India
Email: pranab@cmi.ac.in

DOI: https://doi.org/10.1090/S0002-9939-08-09613-5
Keywords: Pseudo reflections, line bundle.
Received by editor(s): July 5, 2007
Received by editor(s) in revised form: March 4, 2008
Published electronically: September 15, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.