Sandra Di Rocco: Toric polarized fibrations and Cayley polytopes

The geometry of a projective toric variety is encoded in the structure of an associated convex integral polytope. Fibrations between toric varieties, embedded in projective space, are associated to certain fibered polytopes. Fibrations between toric varieties, having a projective space as generic fiber, define polytopes called strict Cayley polytopes. It turns out that this class of polytopes encodes exceptional geometrical properties of the corresponding toric embeddings. Batyrev and Nill have recently conjectured a relation between the degree of a convex polytope and the property of having a Cayley-structure. An overview of the geometrical ideas behind the proof of the conjecture for smooth polytopes will be presented.

This is joint work with A. Dickenstein and R. Piene.