Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbf{ K}$. Let $V_\bullet$ be a graded linear series associated to a line bundle $L$ on $X$, that is, a collection $\{ V_m\}_{ m\in\mathbb{ N}}$ of vector subspaces $V_m\subseteq H^0(X,L^{ \otimes m})$ such that $V_0=\mathbf{ K}$ and $V_k\cdot V_\ell\subseteq V_{ k+\ell}$ for all $k,\ell\in\mathbb{ N}$. For each $m$ in the semigroup \[ \mathbf{ N}(V_\bullet)=\{ m\in\mathbb{ N}\mid V_m\ne 0\},\] the linear series $V_m$ defines a rational map \[ \phi_m\colon X\dashrightarrow Y_m\subseteq\mathbb{ P}(V_m), \] where $Y_m$ denotes the closure of the image $\phi_m(X)$. We show that for all sufficiently large $m\in \mathbf{ N}(V_\bullet)$, these rational maps $\phi_m\colon X\dashrightarrow Y_m$ are birationally equivalent, so in particular $Y_m$ are of the same dimension $\kappa$, and if $\kappa=n$ then $\phi_m\colon X\dashrightarrow Y_m$ are generically finite of the same degree. If $\mathbf{ N}(V_\bullet)\ne\{ 0\}$, we show that the limit \[ \operatorname{ vol}_\kappa(V_\bullet)=\lim_{ m\in \mathbf{ N}(V_\bullet)}\frac{ \dim_\mathbf{ K} V_m}{ m^\kappa/\kappa!}\] exists, and $0<\operatorname{ vol}_\kappa(V_\bullet)<\infty$. Moreover, if $Z\subseteq X$ is a general closed subvariety of dimension $\kappa$, then the limit \[ (V_\bullet^\kappa\cdot Z)_\text{ mov}=\lim_{ m\in \mathbf{ N}(V_\bullet)}\frac{ \#\bigl((D_{ m,1}\cap\cdots\cap D_{ m,\kappa}\cap Z)\setminus \operatorname{ Bs}(V_m)\bigr)}{ m^\kappa}\] exists, where $D_{ m,1},\ldots,D_{ m,\kappa}\in |V_m|$ are general divisors, and \[ (V_\bullet^\kappa\cdot Z)_\text{ mov}=\operatorname{ deg}\bigl(\phi_m|_Z\colon Z\dashrightarrow \phi_m(Z)\bigr)\operatorname{ vol}_\kappa(V_\bullet) \] for all sufficiently large $m\in\mathbf{ N}(V_\bullet)$.
Categories: math.AG