Step 3. By construction, (W, Γw) is klt.
---- klt 往往是 MMP的前奏.
(用于MMP的klt配对, 其相往往出自构造?)
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Run an MMP on Kw + Γw over X and let Y' be the resulting model.
---- MMP 联系着三个空间. 图解:
国w ~> MMP ~> Y'
↓
...............X
套路: 将X回拉到W, 造相得klt配对, 跑MMP得Y'.
.
Since Kw + Γw = G/X, by the negativity lemma, the MMP contracts any component of G with positive coefficient, hence Y' --> X is an isomorphism over the complement of finitely many closed points.
---- “G/X” 标识为“舅”.
---- Kw + Γw = ?/X 标识为“国舅(?)”.
---- “negativity lemma” 参链接. (提及 Lemma 3.39 in the book by Kollar-Mori)
助记: 国舅(G)+负引理 ==> MMP 压缩 [G]₊.
( [G]₊ 是指G的任何正系数分量).
---- 应该跟G exceptional 有关.(?)
---- 后半句: Y' --> X c.c. 同构.
助记: 国舅(G)+负引理 ==> MMP 压缩 [G]₊ ==> Y' --> X c.c. 同构.
.
Moreover, since Kw + Γw ≡ v(B~ + tL~) + F/X and since T is not a component of v(B~ + tL~) + F, the MMP does not contract T.
---- “ v(B~ + tL~) + F/X” 标识为“父”(变格).
----“Kw + Γw ≡ v(B~ + tL~) + F/X”标识为“国父”.
(变格与“国”对接).
---- “v(B~ + tL~) + F” 标识为“父”(本格).
助记: 国父 + T≠[父] ==> MMT 不压缩 T.
.
Let AY' be the pullback of A.
---- “侯”出马了...
.
By boundedness of the length of extremal rays [17] and by the base point free theorem, KY' + ΓY' + 2dAY' is semi-ample, globally.
助记: eR有界 + 基点自由 ==> 2-团(Y')欠丰.
.
评论: Step3 的落点是 KY' + ΓY' + 2dAY' semi-ample.
.
小结: Step3 读写完毕. 命题5.9读写完毕.
(注: Step2之后跳到Step7倒着读了).
---- 大致上, Step3~6都是围绕“丰”展开论述.