Newton–Okounkov bodies of exceptional curve valuations

Abstract

We prove that the Newton–Okounkov body associated to the flag $E_{\bullet }:=\{X=X_r\supset E_r\supset \{q\}\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane ${\mathbb{P}}^2$, with respect to the pull-back of the line-bundle ${\mathcal{O}}_{{\mathbb{P}}^2}(1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton–Okounkov bodies which turn out to be triangular.