Classification of smooth Fano polytopes

By Mikkel Øbro
PhD Dissertations
January 2008

A simplicial lattice polytope containing the origin in the interior is called a smooth Fano polytope, if the vertices of every facet is a basis of the lattice. The study of smooth Fano polytopes is motivated by their connection to toric varieties.

The thesis concerns the classification of smooth Fano polytopes up to isomorphism.

A smooth Fano $d$-polytope can have at most $3d$ vertices. In case of $3d$ vertices an explicit classification is known. The thesis contains the classification in case of $3d-1$ vertices.

Classifications of smooth Fano $d$-polytopes for fixed $d$ exist only for $d\leq 5$. In the thesis an algorithm for the classification of smooth Fano $d$-polytopes for any given $d$ is presented. The algorithm has been implemented and used to obtain the complete classification for $d\leq 8$.

Thesis advisor: Johan P. Hansen
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