Toric Fano varieties and convex polytopes

In this thesis we study toric Fano varieties. Toric varieties are a particular class of algebraic variety which can be described in terms of combinatorial data. Toric Fano varieties correspond to certain convex lattice polytopes whose boundary lattice points are dictated by the singularities involved. Terminal toric Fano varieties correspond to convex lattice polytopes which contain only the origin as an internal lattice point, and whose boundary lattice points are precisely the vertices of the polytope. The situation is similar for canonical toric Fano varieties, with the exception that the condition on boundary lattice points is relaxed. We call these polytopes terminal (or canonical) Fano polytopes. The heart of this thesis is the development of an approach to classifying Fano polytopes, and hence the associated varieties. This is achieved by ordering the polytopes with respect to inclusion. There exists a finite collection of polytopes which are minimal with respect to this ordering. It is then possible to “grow” these minimal polytopes in order to obtain a complete classification. Critical to this method is the ability to find the minimal polytopes. Their description is inductive, requiring an understanding of the lower-dimensional minimal polytopes. A generalisation of weighted projective space plays a crucial role – the associated simplices form the building blocks of the minimal polytopes. A significant part of this thesis is dedicated to attempting to understand these building blocks. A classification of all toric Fano threefolds with at worst terminal singularities is given. The three-dimensional minimal canonical polytopes are also found, making a complete classification possible.