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新入の者--> What is going on ? (redirected)
本期开始改变画风,搭载数学类学院等有用链接。
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如何对付较长的证明?
(接上回+oo) Step 2. 第一段。
Assume that there exist a natural number n divisible by I(R) and a divisor 0 <= GY ~ (n+1)M - n(KY+BY) such that (Y, BY^+: = BY + 1/n GY) is lc near T. Let G be the pushdown of GY. Then 0 <= G ~ (n+1)M - n(Kx + B). We claim that (X, B^+:= B + 1/n G) is lc near S.
评论:此段表明,将从Y的层面出发,以间接方式证明定理的结论。
---- 第一句:以假设的形式,在Y定义的层面上,重述了定理1.7的结论。
---- 第二句:由假设的 GY 得到 G。
---- 第三句:得到 G 的结论。
---- 第四句:声明最终的结论。
加评:此段的实质是第一句的假设(其它的暂时都是“空话”)。简记:
GY T
| / | n|I(R)
(Y)^+ lc
补问:定理1.7的结论是怎么想到的?
---- 也许只能从“调用”此定理的上下文里寻找答案。
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Step 2. 第二段。
Let f: Y --> Z be the contraction defined by the semi-ample divisor MY. By assumption and construction, M|S ~ 0, T maps into S, and f factors through Y --> X. Thus T and S~, the birational transform of S, are both mapped to the same point by f, say z. It is enough to show (Y, BY^+) is lc over z because KY + BY^+ is the pullback of Kx + B^+ and f^{-1}{z} contains the pullback of S under Y --> X.
评论:此段进一步明确了思路,归结到 “to show (Y, BY^+) is lc over z”.
问题:contraction 的定义。
问题:f factors through Y --> X。
问题:pullback 的定义。
加评:此段宜于识记,而不宜硬去理解(比如,semi-ample divisor 可定义出 contraction)。简记:
(Kx)^+<~(KY)^+
MY S<~SY⊂f*{z}
| ↓↓
f M|S~0 , T->S (Y, BY^+) lc
\ ↓ |
Y --> Z ==> T, S~~> z
↓
X
又评:作者只勾勒了关键节点。
---- 或出于某种考虑(突出重点?)
---- 或由于此前有类似证明([3, Pro.6.7])。
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Step 2. 第三段。
Assume (Y, BY^+) is not lc over z. Pick a sufficiently small positive number λ and let ΘY:= λ ΓY + (1-κ)BY^+.Then (Y, ΘY) is plt near T but it is not lc over z, hence the non-klt locus of this pair has at least two connected components near f^{-1}{z}. Moreover, -(KY + ΘY) = -λ (KY + ΓY) - (1-λ)(KY + BY^+) ~Q -λ(KY + ΓY)/Z is ample over Z. We then get a contradiction by the connectedness principle [21, Theorem 7.14].
评论:对上一段归结的命题做反证法。逐句评论:
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Assume (Y, BY^+) is not lc over z.
---- 标准的反证法格式:竖靶子。
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Pick a sufficiently small positive number λ and let ΘY:= λ ΓY + (1-κ)BY^+.
---- ΓY 相当于二阶的T,BY^+相当于二阶的BY,ΘY相当于二阶的ΓY(或三阶的T)。
---- 有点象迭代升级。
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Then (Y, ΘY) is plt near T but it is not lc over z...
---- Y 搭配ΘY(即三阶的T)在T 附近保持 plt.
---- 并受反证假设的“感染”,即“not lc over z”.
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...hence the non-klt locus of this pair has at least two connected components near f^{-1}{z}.
---- 似乎是某种显然的后果,但根据并不清楚。待考。
名词:non-klt locus; connected components.
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Moreover, -(KY + ΘY) = -λ (KY + ΓY) - (1-λ)(KY + BY^+) ~Q -λ(KY + ΓY)/Z is ample over Z.
---- 粉色项似乎被吸收了,但根据并不清楚。待考。
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We then get a contradiction by the connectedness principle [21, Theorem 7.14].
---- 得到矛盾(根据是1992年的一篇文章)。
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Step 2. 第四段。
Now replace (X, B) with (Y, BY) and replace S with T.
---- 这样取代的目的不明确。待考。
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小结:完成 Step 2 读写(难度略大、隐含调用较多)。
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第一轮读写链接(按目录顺序)
Abstract 8/4
Introduction
Boundedness of singular Fano varieties (1) 8/5
Boundedness of singular Fano varieties (2) 8/6
Boundedness of singular Fano varieties (3) 8/7
Boundedness of singular Fano varieties (4) 8/8
Boundedness of singular Fano varieties (5) 8/9
Boundedness of singular Fano varieties (6) 8/9
Jordan property of Cremona groups 8/10
Lc thresholds of lR-linear systems 8/11
Lc thresholds of anti-log canonical systems of Fano pairs (1) 8/12
Lc thresholds of anti-log canonical systems of Fano pairs (2) 8/13
Lc thresholds of R-linear systems with bounded degree 8/14
Complements near a divisor 8/15
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