On endomorphisms of arrangement complements
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- by Şevda Kurul and Annette Werner PDF
- Proc. Amer. Math. Soc. 147 (2019), 2797-2808 Request permission
Abstract:
Let $\Omega$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $\Omega$ extends to an endomorphism of the tropical compactification $X$ of $\Omega$ associated to the Bergman fan structure on the tropical variety $\operatorname {trop}(\Omega )$. This generalizes a result in [Compos. Math. 149 (2013), pp. 1211–1224], which states that every automorphism of Drinfeld’s half-space over a finite field $\mathbb {F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb {F}_q$-rational linear subspaces. This successive blow-up is in fact the minimal wonderful compactification by de Concini and Procesi, which coincides with $X$ by results of Feichtner and Sturmfels. Whereas the proof in [Compos. Math. 149 (2013), pp. 1211–1224] is based on Berkovich analytic geometry over the trivially valued finite ground field, the generalization proved in the present paper relies on matroids and tropical geometry.References
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Additional Information
- Şevda Kurul
- Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany
- Email: kurul@math.uni-frankfurt.de
- Annette Werner
- Affiliation: Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 6-8, 60325 Frankfurt am Main, Germany
- MR Author ID: 612980
- Email: werner@math.uni-frankfurt.de
- Received by editor(s): August 23, 2017
- Received by editor(s) in revised form: September 28, 2018
- Published electronically: March 15, 2019
- Additional Notes: Research on this paper was supported by DFG grant WE-4279/7.
- Communicated by: Lev Borisov
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 2797-2808
- MSC (2010): Primary 14T05, 52C35
- DOI: https://doi.org/10.1090/proc/14468
- MathSciNet review: 3973883