Abstract
In the hierarchy of structural sophistication for lattice polytopes, normal polytopes mark a point of origin; very ample and Koszul polytopes occupy bottom and top spots in this hierarchy, respectively. In this paper we explore a simple construction for lattice polytopes with a twofold aim. On the one hand, we derive an explicit series of very ample 3-dimensional polytopes with arbitrarily large deviation from the normality property, measured via the highest discrepancy degree between the corresponding Hilbert functions and Hilbert polynomials. On the other hand, we describe a large class of Koszul polytopes of arbitrary dimensions, containing many smooth polytopes and extending the previously known class of Nakajima polytopes.
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Acknowledgments
We thank Winfried Bruns and Serkan Hoşten for helpful comments and providing us with an invaluable set of examples of very ample polytopes. We also thank Milena Hering for pointing out the overlap of our work with [12] and two anonymous referees for helpful comments. The last author also thanks Mathematisches Forschungsinstitut Oberwolfach for the great working atmosphere and hosting. This research was partially supported by the U. S. National Science Foundation through the Grants DMS-1162638 (Beck), DGE-0841164 (Delgado), DMS-1000641 & DMS-1301487 (Gubeladze), and the Polish National Science Centre Grant No. 2012/05/D/ST1/01063 (Michałek).
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Beck, M., Delgado, J., Gubeladze, J. et al. Very ample and Koszul segmental fibrations. J Algebr Comb 42, 165–182 (2015). https://doi.org/10.1007/s10801-014-0577-7
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DOI: https://doi.org/10.1007/s10801-014-0577-7