摘要
Let X be a complete variety of dimension n over an algebraically closed field K. Let V be a graded linear series associated to a line bundle L on X, that is, a collection {Vm}m∈N of vector subspaces Vm ⊆ H0(X,L⊗m) such that V0 = K and Vk V ⊆ Vk+ for all k, ∈ N. For each m in the semigroup (equation presented) the linear series Vm defines a rational map (equation presented) where Ym denotes the closure of the image φm(X). We show that for all sufficiently large m ∈ N(V ), these rational maps φm: X -→ Ym are birationally equivalent, so in particular Ym are of the same dimension κ, and if κ = n then φm: X -→ Ym are generically finite of the same degree. If N(V ) ≠ {0}, we show that the limit (equation presented) exists, and 0 < volκ(V ) < ∞. Moreover, if Z ⊆ X is a general closed subvariety of dimension κ, then the limit (equation presented) exists, where Dm,1, . . . ,Dm,κ ∈ |Vm| are general divisors, and (equation presented) for all sufficiently large m ∈ N(V).