Fano varieties with finitely generated semigroups in the Okounkov body construction

The Okounkov body is a construction which, to an effective divisor D on an n-dimensional algebraic variety X, associates a convex body in the n-dimensional Euclidean space R^n. It may be seen as a generalization of the moment polytope of an ample divisor on a toric variety, and it encodes rich numerical information about the divisor D. When constructing the Okounkov body, an intermediate product is a lattice semigroup, which we will call the Okounkov semigroup. Recently it was discovered that finite generation of the Okounkov semigroup has interesting geometric implication for X regarding toric degenerations and integrable systems, however the finite generation condition is difficult to establish except for some special varieties X. In this article, we show that smooth projective del Pezzo varieties have finitely generated Okounkov semigroups, providing the first family of nontrivial higher dimensional examples that are not coming from representation theory. Our result also gives a partial answer to a question of Anderson, Kuronya, and Lozovanu.