Newton-Okounkov bodies of exceptional curve valuations

Abstract

[EN] We prove that the Newton-Okounkov body associated to the flag E-circle := {X = X-r superset of E-r superset of {q}}, defined by the surface X and the exceptional divisor E-r given by any divisorial valuation of the complex projective plane P-2, with respect to the pull-back of the line-bundle O-P2 (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.