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On endomorphisms of arrangement complements


Authors: Şevda Kurul and Annette Werner
Journal: Proc. Amer. Math. Soc. 147 (2019), 2797-2808
MSC (2010): Primary 14T05, 52C35
DOI: https://doi.org/10.1090/proc/14468
Published electronically: March 15, 2019
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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.


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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
Email: werner@math.uni-frankfurt.de

DOI: https://doi.org/10.1090/proc/14468
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
Article copyright: © Copyright 2019 American Mathematical Society