Séminaire Lotharingien de Combinatoire, 89B.62 (2023), 12 pp.

Sofía Garzón and Christian Haase

Fine Polyhedral Adjunction Theory

Abstract. Originally introduced by Fine and Reid in the study of plurigenera of toric hypersurfaces (Fine 1983, Reid et al. 1985), the Fine interior of a lattice polytope got recently into the focus of research: it is has been used for constructing canonical models in the sense of Mori Theory (Batyrev 2020). Based on the Fine interior, we propose here a modification of the original adjoint polytopes by defining the Fine adjoint polytope P^{F(s) of P as consisting of the points in P that have lattice distance at least s to all valid inequalities for P.
We obtain a Fine polyhedral adjunction theory that is, in many respects, better behaved than its original analogue.
Many existing results in polyhedral adjunction theory carry over, some with stronger conclusions, such as decomposing polytopes into Cayley sums, and most with simpler, more natural proofs as in the case of the finiteness of the Fine spectrum.

Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.

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