Numerical and enumerative results on Okounkov bodies

TL;DR

This paper investigates three problems related to Okounkov bodies on projective varieties, including a geometric Fujita approximation, a Jow-type theorem, and cardinality formulas for Minkowski bases on smooth surfaces.

Contribution

It introduces a geometric version of Fujita's approximation, a Jow-type theorem, and cardinality formulas for Minkowski bases, advancing understanding of Okounkov bodies in algebraic geometry.

Findings

Established a geometric Fujita approximation theorem

Proved a Jow-type theorem for Okounkov bodies

Derived cardinality formulas for Minkowski bases on smooth surfaces

Abstract

In this note we focus on three independent problems on Okounkov bodies for projective varieties. The main goal is to present a geometric version of the classical Fujita Approximation Theorem, a Jow-type theorem and a cardinality formulae for Minkowski bases on a certain class of smooth projective surfaces.