finite WLCM WQZ3DNQA

finite WLCM

Theorem E (Finiteness of models). Let π : X −→ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Fix a general ample Q-divisor A ≥ 0 over U . Let V be a finite dimensional affine subspace of WDivR(X) which is defined over the rationals. Suppose that there is a Kawamata log terminal pair (X, ∆0). Then there are finitely many birational maps ψj : X Zj over U , 1 ≤ j ≤ l such that if ψ : X Z is a weak log canonical model of KX + ∆ over U , for some ∆ ∈ LA(V ), then there is an index 1 ≤ j ≤ l and an isomorphism ξ : Zj −→ Z such that ψ = ξ ◦ ψj. (pdf) (Birkar 等, 2009, p. 414)