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

Abstract

Let $\pi :X\to {\mathbb{P}}^2$ 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 $0\le n\le 9$ 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 ${\mathbb{CP}}^2$ and a point on the line in general position. Furthermore, we show what Nagata's Conjecture predicts on these Okounkov bodies when $n>9$.

12 pages, 1 figure

References in corpus (3)

• Convex bodies and algebraic equations on affine varieties
• Functions on Okounkov bodies coming from geometric valuations (with an appendix by Sébastien Boucksom)
• Seshadri constants via Okounkov functions and the Segre-Harbourne-Gimigliano-Hirschowitz Conjecture