# Newton–Okounkov bodies of exceptional curve valuations

### Carlos Galindo

Universitat Jaume I, Castellón de la Plana, Spain### Julio José Moyano-Fernández

Universitat Jaume I, Castellón de la Plana, Spain### Francisco Monserrat

Universitat Politècnica de València, Spain### Matthias Nickel

Universität Frankfurt, Germany

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## 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.

## Cite this article

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. 2147–2182

DOI 10.4171/RMI/1195