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Definition 2.9. [BCHM, Definition 3.6.5] Let π : (X, D) → U be a projective morphism of normal quasi-projective varieties and let D be an R-Cartier divisor on X. Let f : X 99K Y be a birational map over U , then Z is a semiample model for D over U if f is (KX + D)-nonpositive and KY + f∗D is semiample over U . Let g : X 99K Z be a rational map over U , then Z is an ample model for D over U if there is an ample divisor H over U on Z such that if p : W → X and q : W → Z resolves g, then q is a contraction morphism, and we may write p∗D ∼R,U q∗H + E, where E ⩾ 0 and for any B ∈ |p∗D/U |R, then B ⩾ E. Definition 2.10. [BCHM, Definition 3.6.7] Let π : (X, D) → U be a projective morphism of normal quasi-projective varieties, if KX + D is log canonical and f : X 99K Y is a birational contraction, then define: (1) Y is a weak log canonical model for KX + D over U if f is (KX (pdf) (Wang 和 Chen, p. 4)