mmp with scaling BTVBQNHJ

mmp with scaling

To begin with, let (X, B) be a Q-factorial KLT pair and let f : X → S be a projective morphism, a scale is an R-divisor H satisfying the following properties: (1) (X, B + H) is KLT. (2) KX + B + H is relatively nef. The idea is to use H to control the progress of MMP. Starting from (X, B) = (X0, B0), we construct the MMP for (X, B) with scaling of H such that in the n-th step we have a Q-factorial KLT pair (Xn, Bn) satisfying the following properties: (1) (Xn, Bn + tn−1Hn) is KLT. (2) KXn + Bn + tn−1Hn is relatively nef. Here Hn the strict transform of H, and tn is defined as the following threshold: tn = min{t ≥ 0 | KXn + Bn + tHn is relatively nef}. We set t−1 = 1. When n = 0, by assumption, t0 ≤ 1. Assume that KX + B is not relatively nef, then t0 > 0. When n > 0, by construction, KXn + Bn + tn−1Hn is relatively nef, and hence tn ≤ tn−1. The inductive construction of this MMP is as the following. Take n ≥ 0. Assume that we already have (Xn, Bn). If tn = 0, then KXn + Bn is relatively nef and the MMP ends. If tn > 0, then we proceed to the next step by the following lemma. (pdf) (Kawamata 和 Jiang, p. 129)