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ψ 和它的兄弟们...
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[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...
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.
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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.
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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
https://blog.sciencenet.cn/blog-315774-1170912.html
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