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

Abstract

Let $\pi: X \rightarrow \mathbb{P}^2$ be the blow-up of $\mathbb{CP}^2$ in $n$ points $x_i$ in very general position, and let $E_i$ be the exceptional divisor over $x_i$. For $0 \leq n \leq 9$ 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 $\mathbb{CP}^2$ and a point on the line in general position. Furthermore, we show what Nagata's Conjecture predicts on these Okounkov bodies when $n > 9$.