finite LTM KYT5TILG

finite LTM

Lemma 7.1. Assume Theorem Cn and Theorem Dn. Let π : X −→ U be a projective morphism of normal quasi-projective varieties, where X has dimension n. Let V be a finite dimensional affine subspace of WDivR(X), which is defined over the rationals. Fix a general ample Q-divisor A over U . Let C ⊂ LA(V ) be a rational polytope such that if ∆ ∈ C, then KX + ∆ is Kawamata log terminal. Then there are finitely many rational maps φi : X Yi over U , 1 ≤ i ≤ k, with the property that if ∆ ∈ C ∩ EA,π(V ), then there is an index 1 ≤ i ≤ k such that φi is a log terminal model of KX + ∆ over U . (pdf) (Birkar 等, 2009, p. 458)