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Toric Birational Geometry and Applications to Lattice Polytopes

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.