[注：最后一句原文作了改正(之前录入时出现错误).]

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本期开始分组发送邮件，搭载数学类学院等链接。

今日学院：暂无。|| 新闻+ || 符号大全、上下标.|| 常用：↑↓ π ΓΔΛΘΩμφΣ∈ ∉ ∪ ∩ ⊆ ⊇ ⊂ ⊃ ≤ ≥ ⌊ ⌋ ⌈ ⌉ ≠ ⁻⁰ ¹ ² ³ ᵈ ₀ ₁ ₂ ₃ ᵢ .

ψ 和它的兄弟们...

(接前：31 30 29) 命题5.7的证明.

Step4. 第二段 (逐句评论).

Let ψ: V --> X be a log resolution of (X, B) on which T is a divisor.

---- 此处引入了一个新的 log resolution.

---- 图解:  ψ ~ V ~ (X, B) ~ T.

---- 对照: φ~ W ~ (X, Λ) ~ Aw (参 Step2*).

.

Define a boundary ΓV = (1 + t)B~ + (1 - eps/4)ΣEi + (1 - a)T where Ei are the exceptional divisors of ψ other than T, a = a(T, X, B), and ~ denotes birational transform.

---- 这里引入了一个边界 ΓV.

---- 它是怎么构造出来的呢 ?

---- ΣEi 也可以记作 E(ψ).

---- 简单起见，引入“向量”记号：

γ  = (1 + t, 1-eps/4, 1 - a).

O = (B~,  E(ψ),  T).

---- 则： ΓV = γ·O.

(O 的分量对应 BET，联想“打赌”).

{---- 注意，ΓV 是边界，下标提示主集合 V.

---- 意味着有配对 (V, ΓV).

(原作没有明显写出此配对)

---- 这就是为何下文忽然冒出个 Kv + Γv.

} 以上括住的是读了下文后补充的.

.

Let aᵢ = a(Eᵢ, X, B). Since (X, B) is eps-lc and since μTΓV = 1 - a, we have Kv +  Γv = Kv + B~ + Σ(1 - aᵢ)Eᵢ + (1 - a)T + tB~ + Σ(aᵢ - eps/4)Eᵢ = ψ*(Kx + B) + tB~ + F where F:=Σ(aᵢ - eps/4)Eᵢ is effective and exceptional over X and its support does not contain T.

---- 发生变化的部分(粉色) 等于上句紫色部分. 

(上句紫色部分做了补项、拆项，为的是引入 aᵢ).

---- ψ*(Kx + B)=Kv+B~+Σ(1 - aᵢ)Eᵢ+(1-a)T.?

---- 原作是从ψ*(Kx + B) 出发拆项补项 ?

.

On the other hand, if a' = a(T, X, B + tB) and a'ᵢ = a(Eᵢ, X, B + tB), then we can write Kv +  Γv = Kv + (1 + t)B~ +Σ(1 - a'ᵢ)Eᵢ + (1 - a')T + Σ(a'ᵢ - eps/4)Eᵢ + (a' - a)T= ψ*(Kx + (1 + t)B) + G where G:= Σ(a'ᵢ - eps/4)Eᵢ + (a' - a)T is exceptional over X.

---- 为引入a' 和 a'i 做了拆项补项.

---- 按颜色对应到上、上一句.

---- 这里的紫色部分与上一句粉色部分结构相同.

(此处只是用a'ᵢ 替换了 aᵢ).

---- ψ*(Kx + (1 + t)B) = Kv + (1 + t)B~ +Σ(1 - a'ᵢ)Eᵢ + (1 - a')T.? 

.

评论：刚才这两句用到星号算符，出处待考.

.

Moreover, if the image of Eᵢ on X is positive-dimensional for some i, then Eᵢ is a component of G with positive coefficient because (X, B + tB) is eps/2-lc outside finitely many closed points.

---- 暂时不明就里. ? (且看后文是否用到)

.

小结：Step4 读写完毕.

Leonhard Euler  Carl Friedrich Gauss  Grothendieck   

Glossary (AG) 

*

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

....

....

  Proposition 5.2 11/9

...

  Proposition 5.5 11/5