Newton-Okounkov bodies of exceptional curve valuations

Abstract

We prove that the Newton-Okounkov body of the flag $E_{\bullet }:=\left\{X=X_r\supset E_r\supset \{q\}\right\}$, 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.

30 pages, 11 figures. V2: The terminology "infinitely singular valuations" used in the first version has been just replaced with the more customary "exceptional curve valuations" (it is simply a matter of denomination)