Sofía Garzón Mora (FU Berlin) - Spectra of Fine polyhedral adjoints and beyond

At the interplay between discrete and toric geometry lies the Fine interior of a lattice polytope, which got recently into the focus of research. From a polytopal perspective, and based on the Fine interior, we will study a modification of the so-called adjoint polytopes and define the Fine adjoint polytope of a polytope P. This consists of the points in P that have lattice distance at least s to all valid inequalities for P. In this manner, we obtain a Fine Polyhedral Adjunction Theory that is better behaved than its original analogue. Many existing results in Polyhedral Adjunction Theory carry over. In this talk, we will focus mainly on one of our conclusions obtained with simpler, more natural proofs as is the case of the finiteness of the Fine spectrum. Namely, we will introduce the \Q-codegree as an invariant arising in the context of toric geometry, and discuss why it can take only finitely many values for polytopes under certain conditions. Some computational results will also be presented, and, if time permits, a second main result on the Cayley structure and decomposition of polytopes will provide a very useful setting for our invariant.