# A question on Okounkov bodies

Let $$X$$ be an irreducible $$n$$-dimensional projective variety, and $$Y_n\subset Y_{n-1}\subset\dots\subset Y_1\subset X$$ a flag of irreducible subvarieties such that $$Y_i$$ has codimension $$i$$ in $$X$$ and it is smooth at the point $$Y_n$$ for any $$i = 1,\dots,n-1$$.

Let $$D$$ be a Cartier divisor on $$X$$. Given a non-zero section $$s\in H^0(X,D)$$ define $$\nu_1 = \nu_1(s) = ord_{Y_1}(s)$$. Now, if $$\{t=0\}$$ is a local equation for $$D$$ in $$X$$ the section $$s$$ determines a section $$\widetilde{s} = st^{-\nu_1}\in H^0(X,D-\nu_1 Y_1)$$ that is not identically zero. Consider $$\widetilde{s}_{|Y_1}$$ and set $$\nu_2 = \nu_2(s) = ord_{Y_2}(\widetilde{s}_{|Y_1})$$.

Proceeding like this we get a valuation $$\nu:H^0(X,D)\rightarrow \mathbb{Z}^n\cup\{\infty\}$$ given by $$\nu(s) = (\nu_1(s),\nu_2(s),\dots,\nu_n(s))$$.

Consider the semi-group $$\Gamma(D) = \{(k,\nu(s))\:|\: 0\neq s\in H^0(X,kD), k\in\mathbb{Z}_{\geq 0}\}\subset\mathbb{Z}^{n+1}_{\geq 0}$$ and let $$\Sigma(D)\subset\mathbb{R}^{n+1}$$ be the closed convex cone generated by $$\Gamma(D)$$.

The Okounkov body associated to $$D$$ with respect to the fixed flag is $$\Delta(D) = \Sigma(D)\cap (\mathbb{R}^n\times \{1\})$$

How can one compute, at least in simple examples, Okounkov bodies? If $$D$$ is ample is it enough to consider the values of the valuation on a basis of $$H^0(X,D)$$? For instance, if $$X = \mathbb{P}^2$$, the flag is given by a point in a line and $$D$$ is the hyperplane section what is $$\Delta(D)$$? Since $$\mathbb{P}^2$$ is toric it should be just a triangle.

• I’ll try to return to answer the example you gave, but just a brief note: you won’t be able to get away just by working with global sections of some ample $D$, since such a divisor need not have any sections. You can probably take some sufficiently ample multiple of $D$ and then divide. Dec 8 '19 at 23:44