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Invariants of hyperplane groups and vanishing ideals of finite sets of points

Part of: Affine geometry General commutative ring theory

Published online by Cambridge University Press:  16 March 2012

H. E. A. Campbell  and
Jianjun Chuai
H. E. A. Campbell
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (eddy@unb.ca; jchuai@unb.ca)
Jianjun Chuai
Affiliation:
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada (eddy@unb.ca; jchuai@unb.ca)
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Abstract

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We define a hyperplane group to be a finite group generated by reflections fixing a single hyperplane pointwise. Landweber and Stong proved that the invariant ring of a hyperplane group is again a polynomial ring in any characteristic. Recently, Hartmann and Shepler gave a constructive proof of this result. By their algorithm, one can always construct generators that are additive. In this paper, we study hyperplane groups of order a power of a prime p in characteristic p and give a slightly different construction of the generators than Hartmann and Shepler. We then show that such generators have a particular form. Furthermore, we show that if the group is defined by a finite additive subgroup W ⊆ , the vanishing ideal of W is generated by polynomials obtained from a set of generators of the invariant ring that are additive. Finally, we give a shorter proof of the fact that the module of the invariant differential 1-forms is free in our situation.

Keywords

hyperplane groupinvariant ringvanishing idealinvariant differential 1-form

MSC classification

Secondary: 13A50: Actions of groups on commutative rings; invariant theory 14R99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 355 - 367
Copyright
Copyright © Edinburgh Mathematical Society 2012

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