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$ 4$ planes in $ {\mathbb{R}}^4$


Author: E. Batzies
Journal: Proc. Amer. Math. Soc. 135 (2007), 3341-3347
MSC (2000): Primary 52C35, 32S22; Secondary 58D29
DOI: https://doi.org/10.1090/S0002-9939-07-08186-5
Published electronically: June 19, 2007
MathSciNet review: 2322766
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Abstract: We establish a homeomorphism between the moduli space $ A_{4,k}^{\rm ord}(\mathbb{R})$ of ordered $ k$-tuples $ (H_1,\ldots ,H_k)$ of 2-dimensional linear subspaces $ H_i \subset \mathbb{R}^4$ (mod $ {\rm GL}_4(\mathbb{R})$) and the quotient by simultaneous conjugation of a certain open subset $ ({\rm GL}_2^{k-3})^* \subset ({\rm GL}_2(\mathbb{R}))^{k-3}$. For $ k=4$, this leads to an explicit computation of the moduli space $ A_{4,4}(\mathbb{R})$ of central 2-arrangements in $ \mathbb{R}^4$ mod $ {\rm GL}_4(\mathbb{R})$ and its subspace $ A_{2,4}({\mathbb{C}})$ of those classes that contain a complex hyperplane arrangement.


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Additional Information

E. Batzies
Affiliation: Fachbereich Mathematik und Informatik, Universität Marburg, 35032 Marburg, Germany
Email: batzies@web.de

DOI: https://doi.org/10.1090/S0002-9939-07-08186-5
Keywords: Arrangements
Received by editor(s): July 27, 2001
Received by editor(s) in revised form: January 23, 2005
Published electronically: June 19, 2007
Dedicated: This paper is dedicated to Julia.
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2007 American Mathematical Society

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