Toric varieties are geometric objects defined combinatorially, much like CW complexes in topology. They can be used to study geometrically combinatorial data like lattice points in convex sets. In Algebraic Geometry, toric varieties are a rich source of examples, a good testing ground for gaining intuition for the general theory.

This course tours several topics of birational geometry, in the special case of toric varieties or general hypersurfaces in toric varieties. We will discuss the classification of singularities and varieties of K-pure type (Fano, Calabi-Yau, canonically polarized). If time permits, we also discuss Calabi-Yau fibrations, the canonical bundle formula or the stability of polarized varieties.

We assume basic knowledge of Birational Geometry and Toric Varieties. Depending on the audiance, we will first recall the basics of convexity and toric varieties.

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