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arXiv · 1010.3887

Few smooth d-polytopes with n lattice points

Tristram Bogart·Christian Haase·Milena Hering·Benjamin Lorenz·Benjamin Nill·Andreas Paffenholz·Günter Rote·Francisco Santos·Hal Schenck
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Abstract

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.

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Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Günter Rote, Francisco Santos, Hal Schenck. 2013-07-17. Few smooth d-polytopes with n lattice points. https://doi.org/10.1007/s11856-015-1175-7

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