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.

  • $\begingroup$ 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. $\endgroup$ Dec 8 '19 at 23:44

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