Iterative dissection of Okounkov bodies of graded linear series on ${\mathbb{C}\mathbb{P}}^2$

Let $\pi :X\to {\mathbb{P}}^2$ be the blow-up of ${\mathbb{C}\mathbb{P}}^2$ in $n$ points $x_i$ in very general position, and let $E_i$ be the exceptional divisor over $x_i$. For $0\le n\le 9$ we calculate Okounkov bodies of graded linear series given by sections of multiples of line bundles $\pi^\ast $\mathcal{O}$_{$\mathbb{P}$^2}(d) \otimes $\mathcal{O}$X(-m\sum{i=1}^n E_i)$ with respect to a flag consisting of a line on ${\mathbb{C}\mathbb{P}}^2$ and a point on the line in general position. Furthermore, we show what Nagata’s Conjecture predicts on these Okounkov bodies when $n>9$.