On Minkowski decomposition of Okounkov bodies on a Del Pezzo surface

Abstract

We show that on a blow up of $ℙ^2$ in 3 general points there exists a finite set of nef divisors $P_1 ,..., P_s$ such that the Okounkov body $∆(D)$ of an arbitrary effective $ℝ–divisor$ $D$ on $X$ is the Minkowski sum \[∆(D) = \sum_{i=1}^sa_i∆(P_i)\] (1) with non-negative coefficients $a_i ∊ ℝ≥0$.