PomeDefinition 17.14.8.D6CC0BBA6D35 Let $L$ be a big line bundle. We write
\[ R(X,L)=\bigoplus _{k=0}^{\infty } \mathrm{H}^0(X,L^ k) \]
for the section ring1 of $L$.
A filtration on $R(X,L)$ is a decreasing family of graded linear subspaces $(\mathcal{F}^{\lambda })_{\lambda \in \mathbb {R}}$ of $R(X,L)$ with graded pieces
\[ \mathcal{F}^{\lambda }=\bigoplus _{k=0}^{\infty } \mathcal{F}^{\lambda }_ k, \]
such that the following conditions are satisfied:
The filtration is left-continuous: For any $\lambda \in \mathbb {R}$, we have
\[ \mathcal{F}^{\lambda }=\bigcap _{\lambda '<\lambda }\mathcal{F}^{\lambda '}; \]the filtration is multiplicative: For any $\lambda ,\lambda '\in \mathbb {R}$ and any $k,k'\in \mathbb {N}$, we have
\[ \mathcal{F}^{\lambda }_ k\cdot \mathcal{F}^{\lambda '}_{k'}\subseteq \mathcal{F}^{\lambda +\lambda '}_{k+k'}; \]there is an integer $C>0$ such that
F76F341D25A817.14.9\begin{equation} \mathcal{F}^{Cm}_{m}=0,\quad \mathcal{F}^{-Cm}_ m=\mathrm{H}^0(X,L^ m) \end{equation}for all $m\in \mathbb {Z}_{>0}$.
Given a filtration $\mathcal{F}$ on $R(X,L)$, we define
\[ \tau _ k(\mathcal{F})=\sup \left\{ \lambda \in \mathbb {R}: \mathcal{F}^{\lambda }_ k\neq 0\right\} . \]
By Fekete's lemma, we can introduce
\[ \tau (\mathcal{F})=\lim _{k\to \infty }\frac{1}{k}\tau _ k(\mathcal{F})=\sup _{k\in \mathbb {Z}_{>0}}\frac{1}{k}\tau _ k(\mathcal{F}). \]
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