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

TL;DR

This paper computes Okounkov bodies for certain linear series on blow-ups of the projective plane with up to nine points, and explores Nagata's Conjecture predictions for more points.

Contribution

It provides explicit calculations of Okounkov bodies for linear series on blow-ups of  with up to nine points and discusses implications of Nagata's Conjecture for more points.

Findings

Explicit Okounkov bodies for n  9 points.

Predictions of Nagata's Conjecture for n > 9.

Insights into linear series on blown-up projective planes.

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 $n > 9$.