Newton–Okounkov bodies of exceptional curve valuations

Abstract

We prove that the Newton–Okounkov body associated to the flag E∙:={X=Xr⊃Er⊃{q}}, defined by the surface X and the exceptional divisor Er given by any divisorial valuation of the complex projective plane P2, with respect to the pull-back of the line-bundle OP2(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.