Okounkov bodies and giansiracusa-giansiracusa valuations

Abstract

Okounkov bodies are convex sets associated with divisors on algebraic varieties that allow for the study of asymptotic properties of both the divisor and the variety. This theory, developed independently by Lazarsfeld–Mustata [5], and Kaveh–Khovanskii [4], is based on tools from discrete semi-groups and valuations associated with divisors. It can be seen as a generalization of the Newton polytope in the context of projective toric varieties. On the other hand, prevaluations, also known by their author’s names as Giansiracusa–Giansiracusa valuations [16], are a more relaxed notion of valuation, replacing the value group with an idempotent semi-ring. This generalization allows for the definition of a natural product of prevaluations, which opens new paths for reinterpreting and/or extending the construction of Okounkov bodies. We will discuss the fundamental concepts and present some preliminary developments towards developing a theory of Okounkov bodies with respect to prevaluations.