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Diagonal splittings of toric varieties and unimodularity


Authors: Jed Chou, Milena Hering, Sam Payne, Rebecca Tramel and Ben Whitney
Journal: Proc. Amer. Math. Soc. 146 (2018), 1911-1920
MSC (2010): Primary 14M25, 52B20; Secondary 13A35, 90C10
DOI: https://doi.org/10.1090/proc/13902
Published electronically: December 7, 2017
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Abstract: We use a polyhedral criterion for the existence of diagonal splittings to investigate which toric varieties $ X$ are diagonally split. Our results are stated in terms of the vector configuration given by primitive generators of the 1-dimensional cones in the fan defining $ X$. We show, in particular, that $ X$ is diagonally split at all $ q$ if and only if this configuration is unimodular, and that $ X$ is not diagonally split at any $ q$ if this configuration is not $ 2$-regular. We also study implications for the possibilities for the set of $ q$ at which a toric variety $ X$ is diagonally split.


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Additional Information

Jed Chou
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: jedchou1@illinois.edu

Milena Hering
Affiliation: School of Mathematics, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingdom
Email: m.hering@ed.ac.uk

Sam Payne
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
Email: sam.payne@yale.edu

Rebecca Tramel
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: rtramel@illinois.edu

Ben Whitney
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: ben_whitney@brown.edu

DOI: https://doi.org/10.1090/proc/13902
Received by editor(s): January 6, 2017
Received by editor(s) in revised form: July 4, 2017
Published electronically: December 7, 2017
Additional Notes: Portions of this research were carried out during an REU project supported under NSF grant DMS-1001859.
The work of the second author was partially supported by EPSRC first grant EP/K041002/1.
The work of the third author was partially supported by NSF CAREER DMS-1149054.
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society

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