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 \[ ϕ_m\colon X\dashrightarrow Y_m\subseteq\mathbb{P}(V_m), \] where $Y_m$ denotes the closure of the image $ϕ_m(X)$. We show that for all sufficiently large $m\in \mathbf{N}(V_\bullet)$, these rational maps $ϕ_m\colon X\dashrightarrow Y_m$ are birationally equivalent, so in particular $Y_m$ are of the same dimension $κ$, and if $κ=n$ then $ϕ_m\colon X\dashrightarrow Y_m$ are generically finite of the same degree. If $\mathbf{N}(V_\bullet)\ne\{0\}$, we show that the limit \[
\operatorname{vol}_κ(V_\bullet)=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\dim_\mathbf{K} V_m}{m^κ/κ!}\] exists, and $0<\operatorname{vol}_κ(V_\bullet)<\infty$. Moreover, if $Z\subseteq X$ is a general closed subvariety of dimension $κ$, then the limit \[
(V_\bullet^κ\cdot Z)_\text{mov}=\lim_{m\in \mathbf{N}(V_\bullet)}\frac{\#\bigl((D_{m,1}\cap\cdots\cap D_{m,κ}\cap Z)\setminus \operatorname{Bs}(V_m)\bigr)}{m^κ}\] exists, where $D_{m,1},\ldots,D_{m,κ}\in |V_m|$ are general divisors, and \[ (V_\bullet^κ\cdot Z)_\text{mov}=\operatorname{deg}\bigl(ϕ_m|_Z\colon Z\dashrightarrow ϕ_m(Z)\bigr)\operatorname{vol}_κ(V_\bullet) \] for all sufficiently large $m\in\mathbf{N}(V_\bullet)$.
Duality for asymptotic invariants of graded families
University of South Alabama（南阿拉巴马大学）；Tulane University（杜兰大学）；University of Education, Hue University（顺化师范大学）；University of Nebraska-Lincoln（内布拉斯加大学林肯分校）
