We prove that the Newton–Okounkov body associated to the flag , defined by the surface and the exceptional divisor given by any divisorial valuation of the complex projective plane , with respect to the pull-back of the line-bundle 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.
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Carlos Galindo, Julio José Moyano-Fernández, Francisco Monserrat, Matthias Nickel, Newton–Okounkov bodies of exceptional curve valuations. Rev. Mat. Iberoam. 36 (2020), no. 7, pp. 2147–2182DOI 10.4171/RMI/1195