PomeDefinition 19.2.4.B62535F66376 Given $S\in \mathcal{S}'(C)$, its Okounkov body is defined by
\[ \Delta (S)\coloneqq \left\{ x\in \mathbb {R}^ n:(x,1)\in C(S) \right\} . \]
19.2 Almost semigroups1ED20E6196A2
1ED20E6196A2
We give a brief presentation of the theory of almost semigroups. The arguments are provided in section 19.7.
Fix an integer $n\geq 0$. Fix a closed convex cone $C\subseteq \mathbb {R}^{n}\times \mathbb {R}_{\geq 0}$ such that $C\cap \{ x_{n+1}=0\} =\{ 0\} $. Here $x_{n+1}$ is the last coordinate of $\mathbb {R}^{n+1}$.
Write $\hat{\mathcal{S}}(C)$ for the set of subsets of $C\cap \mathbb {Z}^{n+1}$ and $\mathcal{S}(C)$ for the set of subsemigroups $S\subseteq C\cap \mathbb {Z}^{n+1}$. For $S\in \hat{\mathcal{S}}(C)$ and each $k\in \mathbb {N}$ such that $S_ k\neq \varnothing $, we write
\[ S_ k\coloneqq \left\{ x\in \mathbb {Z}^ n: (x,k)\in S \right\} . \]
Note that $S_ k$ is a finite set by our assumption on $C$.
We introduce a pseudometric on $\hat{\mathcal{S}}(C)$ as follows:
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19.2.1
\begin{equation} d_{\mathrm{sg}}(S,S')\coloneqq \varlimsup _{k\to \infty } k^{-n} \left(|S_ k|+|S'_ k|-2|(S\cap S')_ k|\right). \end{equation}
Here $|\bullet |$ denotes the cardinality of a finite set. The quantity $d_{\mathrm{sg}}$ defined above is a pseudometric on $\hat{\mathcal{S}}(C)$. Given $S,S'\in \hat{\mathcal{S}}(C)$, we say $S$ is equivalent to $S'$ and write $S\sim S'$ if $d_{\mathrm{sg}}(S,S')=0$. This is an equivalence relation.
The volume of $S\in \mathcal{S}(C)$ is defined as
\[ \operatorname{vol}S\coloneqq \lim _{k\to \infty } (ka)^{-n}|S_{ka}|=\varlimsup _{k\to \infty } k^{-n}|S_ k|, \]
where $a$ is a sufficiently divisible positive integer. The existence of the limit and its independence from $a$ both follow from theorem 19.7.20.
We define $\overline{\mathcal{S}}(C)$ as the closure of $\mathcal{S}(C)$ in $\hat{\mathcal{S}}(C)$ with respect to the topology defined by the pseudometric $d_{\mathrm{sg}}$. The function $\operatorname{vol}\colon \mathcal{S}(C)\rightarrow \mathbb {R}$ admits a unique $1$-Lipschitz extension to
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19.2.2
\begin{equation} \operatorname{vol}\colon \overline{\mathcal{S}}(C)\rightarrow \mathbb {R}. \end{equation}
Given $S\in \hat{\mathcal{S}}(C)$, we will write $C(S)\subseteq C$ for the closed convex cone generated by $S\cup \{ 0\} $. Moreover, for each $k\in \mathbb {Z}_{>0}$ such that $S_ k\neq \varnothing $, we define
\[ \Delta _ k(S)\coloneqq \mathrm{Conv}\left\{ k^{-1}x\in \mathbb {R}^ n:x\in S_ k \right\} \subseteq \mathbb {R}^ n. \]
Here $\mathrm{Conv}$ denotes the convex hull.
Note that for any $S\in \mathcal{S}'(C)$, the cone $C(S)$ has full dimension (i.e., the topological interior is non-empty). Given a full-dimensional subcone $C'\subseteq C$, we have $C'\cap \mathbb {Z}^{n+1}\in \mathcal{S}'(C)$. We recall the following definition from [KK12].
Definition 19.2.9.78B565EC56CF We define $\overline{\mathcal{S}'(C)}_{>0}$ as the set of elements in the closure of $\mathcal{S}'(C)$ in $\hat{\mathcal{S}}(C)$ with positive volume. An element in $\overline{\mathcal{S}'(C)}_{>0}$ is called an almost semigroup in $C$.
Theorem 19.2.10.2534E787147E The Okounkov body map $\Delta \colon \mathcal{S}'(C) \rightarrow \mathcal{K}_ n$ as defined in definition 19.2.4 admits a unique continuous extension
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19.2.11
\begin{equation} \Delta \colon \overline{\mathcal{S}'(C)}_{>0}\rightarrow \mathcal{K}_ n. \end{equation}
Moreover, for any $S\in \overline{\mathcal{S}'(C)}_{>0}$, we have
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19.2.12
\begin{equation} \operatorname{vol}S=\operatorname{vol}\Delta (S). \end{equation}
Definition 19.2.3.923345ACA44A Let $\mathcal{S}'(C)$ be the subset of $\mathcal{S}(C)$ consisting of semigroups $S$ such that $S$ generates $\mathbb {Z}^{n+1}$ (as an Abelian group).
Theorem 19.2.5.37D86D761E81 For each $S\in \mathcal{S}'(C)$, we have
Moreover, as $k\to \infty $,
Proof. The equalities (19.2.6) follow from theorem 19.7.20.
It remains to prove (19.2.7). By the argument of [Lemma 4.8, WN14], for any compact set $K\subseteq \operatorname{Int}\Delta (S)$, there is $k_0>0$ such that for any $k\geq k_0$, $\alpha \in K\cap (k^{-1}\mathbb {Z})^ n$ implies that $\alpha \in \Delta _ k(S)$.
To conclude, fix $\epsilon >0$. Choose $\delta >0$ such that $K\coloneqq \Delta (S)_{\delta }$ has positive volume and $d_{\mathrm{Haus}}(K,\Delta (S))<\epsilon /2$. Applying 19.7.7 to $K$, we find that $\Delta _ k(S)$ contains $K_{n^{1/2}k^{-1}}$ for all large $k$. After increasing $k$, we may also assume that $d_{\mathrm{Haus}}(K,K_{n^{1/2}k^{-1}})<\epsilon /2$. Since $\Delta _ k(S)\subseteq \Delta (S)$, this implies (19.2.7). $\square$
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19.2.6
\begin{equation} \operatorname{vol}S=\lim _{k\to \infty } k^{-n}|S_ k|=\operatorname{vol}\Delta (S)>0. \end{equation}
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19.2.7
\begin{equation} \Delta _{k}(S)\xrightarrow {d_{\mathrm{Haus}}}\Delta (S). \end{equation}
Corollary 19.2.8.9FB3E8CE18F7 Let $S,S'\in \mathcal{S}'(C)$. Assume that $\operatorname{vol}(S\cap S')>0$. Then we have
\[ d_{\mathrm{sg}}(S,S')=\operatorname{vol}(S)+\operatorname{vol}(S')-2\operatorname{vol}(S\cap S'). \]
Proof. This is a direct consequence of 19.7.25 and (19.2.6). $\square$
Corollary 19.2.13.511230BE9AE5 Suppose that $S,S'\in \overline{\mathcal{S}'(C)}_{>0}$ with $S\subseteq S'$. Then
Proof. Let $S^ j,S'^ j\in \mathcal{S}'(C)$ be elements such that $S^ j\to S$, $S'^ j\to S'$. Then it follows from 19.7.19 that $S^ j\cap S'^ j \to S$. Since $\operatorname{vol}$ is continuous, for large $j$, $S^ j\cap S'^ j$ has positive volume and hence lies in $\mathcal{S}'(C)$ by 19.7.25. We may therefore replace $S^ j$ by $S^ j\cap S'^ j$ and assume that $S^ j\subseteq S'^ j$. Hence, (19.2.14) follows from the continuity of $\Delta $ proved in theorem 19.7.32. $\square$
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