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.

中文翻译：

---

牛顿-奥孔科夫曲线异常估值机构

我们证明了与标志$ E _ {\ bullet}：= \ {X = X_r \ supset E_r \ supset \ {q \} \} $相关的牛顿-奥孔科夫体，由曲面$ X $和例外除数定义相对于线束$ \ mathcal {O} _ {\ mathbb {P} ^ 2的拉回，由复投影平面$ \ mathbb {P} ^ 2 $的除数估值得到的$ E_r $ }（1）$是三角形或四边形，表示它是三角形还是四边形。我们还描述了该图的顶点。最后，我们引入了一个大的标志族，我们可以明确地确定它们的牛顿-奥孔科夫体，它们最终是三角形的。