Proposition 20 ([An exact presentation]). Fix an integer d≥1 and a finite set Φ⊂[0,1]∩ . There is an integer p₀=p₀(d,Φ)>0 , clearing the denominators of Φ , with the following property. Let f:X→Z be a contraction of normal projective …

Put K=(Z) and let F=X×_Z K be the generic fibre. Since f is a contraction and the characteristic is zero, F is geometrically integral. It is normal, its induced pair (F,B_F) is geometrically lc, and K_F+B_F0 . These assertions may be checked on a log resolution: after shrinking the base, the resolution and its marked strata have the required generic smoothness, and the discrepancy formula restricts to the fibres. We choose the canonical divisor on F by dividing a rational top form on X by a rational top form on Z . Theorem 2.1 ([Normal log canonical indices [Theorem~1.2]), applied to the geometric generic fibre, gives a principal multiple in a degree depending only on its dimension and Φ . Taking a common multiple over dimensions at most d , and also clearing Φ , gives p_0 . The theorem applies over the algebraic closure of K by characteristic-zero comparison: all data descend to an algebraically closed field admitting an embedding into . By Lemma 1 ([Descent in the same degree]), this trivialization descends to K in the same degree. Consequently we may choose ψ∈(X)^* such that V:=p_0(K_X+B)+(ψ) has no component dominating Z . It remains to show that this particular vertical divisor is a pullback. Choose n>0 and h∈(X)^* with n(K_X+B-f^*L)=(h) . Then n(V-p_0f^*L)=(h^ p_0 ψ^n). The divisor on the right is vertical. Its defining function has zero divisor on the normal projective generic fibre and hence belongs to K^* , because H^0(F,_F)=K . Write h^ p_0 ψ^n=f^*a for a∈K^* . We obtain V=f^*(p_0L+1n(a)). Thus D_Z=L+(np_0)^ -1 (a) has all the asserted properties.