# Newton-Okounkov bodies of exceptional curve valuations

## TL;DR

This paper characterizes the shape of Newton-Okounkov bodies for certain valuations on the complex projective plane, showing they are either triangles or quadrilaterals and explicitly describing their vertices.

## Contribution

It provides a complete classification of Newton-Okounkov bodies for divisorial valuations on , identifying when they are triangles or quadrilaterals and explicitly describing their vertices.

## Key findings

- Newton-Okounkov bodies are triangles or quadrilaterals.
- Explicit vertices of these bodies are described.
- A large family of flags with triangular Newton-Okounkov bodies is identified.

## 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.