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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on projective normality


Authors: Huah Chu, Shou-Jen Hu and Ming-chang Kang
Journal: Proc. Amer. Math. Soc. 139 (2011), 1989-1992
MSC (2010): Primary 13A02, 13A50, 14Lxx
Published electronically: November 15, 2010
MathSciNet review: 2775375
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Abstract: Let $ G$ be any finite group, $ G\to GL(V)$ be a representation of $ G$, where $ V$ is a finite-dimensional vector space over an algebraically closed field $ k$. Theorem. Assume that either $ char k = 0$ or $ char k = p > 0$ with $ p \nmid \vert G\vert$. Then the quotient variety $ \mathbb{P}(V)/G$ is projectively normal with respect to the line bundle $ \mathcal{L}$, where $ \mathcal{L}$ is the descent of $ \mathcal{O}(1)^{\otimes m}$ from $ \mathbb{P}(V)$ with $ m=\vert G\vert!$. This partially solves a question raised in the paper of Kannan, Pattanayak and Sardar [Proc. Amer. Math. Soc. 137 (2009), 863-867].


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Additional Information

Huah Chu
Affiliation: Department of Mathematics, National Taiwan University, Taipei, Taiwan
Email: hchu@math.ntu.edu.tw

Shou-Jen Hu
Affiliation: Department of Mathematics, Tamkang University, Taipei, Taiwan
Email: sjhu@math.tku.edu.tw

Ming-chang Kang
Affiliation: Department of Mathematics and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan
Email: kang@math.ntu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10777-3
PII: S 0002-9939(2010)10777-3
Keywords: Projectively normal, rings of invariants, graded algebras
Received by editor(s): December 6, 2009
Received by editor(s) in revised form: June 4, 2010
Published electronically: November 15, 2010
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.