For toric varieties, adjunction theory of polarized algebraic varieties amounts to "moving inwards" all facets of a lattice polytope at the same speed. Inspired by what is nowadays called the Fine interior [Batyrev], we propose a "Fine adjunction theory", moving inwards all valid inequalities at the same speed. In many respects, we obtain a better behaved theory.
In this talk, I will focus mainly on the finiteness of the Fine Q-codegree spectrum. A finiteness conjecture by Fujita for the classical analogue was proved only recently by Di Cerbo. For the Fine version, finiteness is easier and does not need smoothness. We try and describe the possible values in small dimensions.
This is joint work with Sofía Garzón.