Asymptotic constructions and invariants of graded linear series

Abstract:

Let $X$ be a complete variety of dimension $n$ over an algebraically closed field $\mathbb {K}$. 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 $V_0=\mathbb {K}$ 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 \[ \phi _m\colon X\dashrightarrow Y_m\subseteq \mathbf {P}(V_m), \] 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 \[ vol_\kappa (V_\bullet )=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\dim _\mathbb {K} V_m}{m^\kappa /\kappa !}\] exists, and $0<vol_\kappa (V_\bullet )<\infty$. Moreover, if $Z\subseteq X$ is a general closed subvariety of dimension $\kappa$, then the limit \[ (V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\lim _{m\in \mathbf {N}(V_\bullet )}\frac {\#\bigl ((D_{m,1}\cap \cdots \cap D_{m,\kappa }\cap Z)\setminus Bs(V_m)\bigr )}{m^\kappa }\] exists, where $D_{m,1},\ldots ,D_{m,\kappa }\in |V_m|$ are general divisors, and \[ (V_\bullet ^\kappa \cdot Z)_{\mathrm {mov}}=\deg \bigl (\phi _m|_Z\colon Z\dashrightarrow \phi _m(Z)\bigr )vol_\kappa (V_\bullet ) \] for all sufficiently large $m\in \mathbf {N}(V_\bullet )$.