Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Günter Rote, Francisco Santos, and Hal Schenck:

Finitely many smooth d-polytopes with n lattice points

Israel Journal of Mathematics 207 (2015), 301–329. doi:10.1007/s11856-015-1175-7, arXiv:1010.3887 [math.CO].

Abstract

We prove that for fixed d and n, there are only finitely many smooth d-dimensional polytopes which contain n lattice points. As a consequence in algebraic geometry, we obtain that for fixed n, there are only finitely many embeddings of Q-factorial toric varieties into n-dimensional projective space Pn that are induced by a complete linear system. We also enumerate all smooth 3-polytopes with at most 12 lattice points.

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