Asymptotic constructions and invariants of graded linear series

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 $V_k\cdot V_{\mathrm{\ell }}\subseteq V_{k+\mathrm{\ell }}$ for all $k,\mathrm{\ell }\in \mathbb{N}$. For each $m$ in the semigroup $$\mathbf{N}(V_{\bullet })=\{m\in \mathbb{N}\mid V_m\mathrm{\ne }0\},$$ the linear series $V_m$ defines a rational map $${\varphi }_m\text{\hspace{0.17em}⁣}:X⇢Y_m\subseteq \mathbb{P}(V_m),$$ where $Y_m$ denotes the closure of the image ${\varphi }_m(X)$. We show that for all sufficiently large $m\in $\mathbf{N}$(V_\bullet)$,theserationalmaps$\phi_m\colon X\dashrightarrow Y_m$arebirationallyequivalent,soinparticular$Y_m$areofthesamedimension$\kappa$,andif$\kappa=n$then$\phi_m\colon X\dashrightarrow Y_m$ are generically finite of the same degree. If $\mathbf{N}(V_{\bullet })\mathrm{\ne }\{0\}$, we show that the limit $${\mathrm{vol}}_{\kappa }(V_{\bullet })=\underset{m\in \mathbf{N}(V_{\bullet })}{\lim }\frac{{\dim }_{\mathbf{K}}{V}_m}{m^{\kappa }\mathrm{/}\kappa !}$$ exists, and $0<{\mathrm{vol}}_{\kappa }(V_{\bullet })<\mathrm{\infty }$. Moreover, if $Z\subseteq X$isageneralclosedsubvarietyofdimension$\kappa$, then the limit $$(V_{\bullet }^{\kappa }\cdot Z{)}_{\text{mov}}=\underset{m\in \mathbf{N}(V_{\bullet })}{\lim }\frac{\mathrm{\#}((D_{m,1}\cap \cdots \cap D_{m,\kappa }\cap Z)\setminus \mathrm{Bs}(V_m))}{m^{\kappa }}$$ exists, where $D_{m,1},\dots ,D_{m,\kappa }\in \mathrm{\mid }{V}_m\mathrm{\mid }$ are general divisors, and $$(V_{\bullet }^{\kappa }\cdot Z{)}_{\text{mov}}=\mathrm{deg}({\varphi }_m{\mathrm{\mid }}_Z\text{\hspace{0.17em}⁣}:Z⇢{\varphi }_m(Z)){\mathrm{vol}}_{\kappa }(V_{\bullet })$$ for all sufficiently large $m\in \mathbf{N}(V_{\bullet })$.