Non-Normal Very Ample Polytopes – Constructions and Examples

ABSTRACT

We answer several questions posed by Beck, Cox, Delgado, Gubeladze, Haase, Hibi, Higashitani, and Maclagan in [Cox et al. 14, Question 3.5 (1),(2), Question 3.6], [Beck et al. 15, Conjecture 3.5(a),(b)], and [Hasse et al. 07, Open question 3 (a),(b) p. 2310, Question p. 2316] by constructing a new family of non-normal very ample polytopes. These polytopes are certain segmental fibrations of unimodular graph polytopes, we explicitly compute their invariants – Hilbert function, Ehrhart polynomial, and gap vector.

KEYWORDS:

• normal polytope
• very ample polytope
• graph polytope
• Hilbert basis
• gap vector
• segmental fibration

2000 AMS SUBJECT CLASSIFICATION:

• 52B20
• 14M25
• 05E40

Acknowledgments

We would like to thank Professor Winfried Bruns for help with computations.

Funding

The research was supported by Polish National Science Centre Grant No. 2012/05/D/ST1/01063 and by Swiss National Science Foundation Grants 200020-144531 and 200021-137574. The article was completed during Michał Lasoń’s stay at Freie Universität Berlin in the frame of Polish Ministry “Mobilność Plus” program.