Toric Birational Geometry and Applications to Lattice Polytopes

TL;DR

This paper investigates the relationship between toric geometry and lattice polytopes, classifying smooth polytopes of small degree and analyzing cones of cycles to aid in birational classification of toric varieties.

Contribution

It extends the classification of smooth lattice polytopes of small degree using techniques from Adjunction and Mori Theory, and describes extremal rays of cones of cycles via fan combinatorics.

Findings

Classification of smooth lattice polytopes with small degree.

Description of extremal rays of cones of cycles in toric varieties.

Analysis of the cone of moving curves and its role in birational classification.

Abstract

Toric geometry provides a bridge between the theory of polytopes and algebraic geometry: one can associate to each lattice polytope a polarized toric variety. In this thesis we explore this correspondence to classify smooth lattice polytopes having small degree, extending a classification provided by Dickenstein, Di Rocco and Piene. Our approach consists in interpreting the degree of a polytope as a geometric invariant of the corresponding polarized variety, and then applying techniques from Adjunction Theory and Mori Theory. In the opposite direction, we use the combinatorics of fans to describe the extremal rays of several cones of cycles of a projective toric variety X. One of these cones is the cone of moving curves, that is the closure of the cone generated by classes of curves moving in a family that sweeps out X. This cone plays an important role in the problem of birational classification of projective varieties.