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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#) Step 3. 第一段。
In view of Steps 1-2, we can assume that there is a boundary Γ ≤ B such that (X, Γ) is plt, \Γ/=S, and A:= αM - (Kx +Γ) is ample for some α∈(0,1). However, M - (Kx + B) may no longer be ample but it is still nef and big.
评论:这里把之前的推导结果作为假设对待。
---- 似乎只涉及到Step1,看不出Step2体现在哪里。
---- 在当前假设下,“M - (Kx + B) is ample” 可以不成立。
---- 这种“倒置”的做法令人略感困惑。
---- 这里定义的 A 要到 Step 6 才出现。
加评:“we can assume” 出现不止一次,或许运用了某种方法。
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Step 3. 第二段。
Let Ks + Bs = (Kx + B)|s. The coefficients of Bs belong to Φ(G) for some finite set G ⊂ [0, 1] of rational numbers containing R and depending only on R [26, Proposition 3.8] [3, Lemma 3.3]. Since (X, Γ) is plt and -(Kx + Γ)|s ~R (α M - (Kx + Γ))|s is ample, S is Fano type. Thus as -(Ks + Bs) ~ (M - (Kx + B))|s is nef, by Theorem 2.13, there is a natural number n divisible by I(G) and depending only on d, G such that Ks + Bs has an n-complement Ks + Bs^+ with Bs^+≥ Bs.
评论:此段处理 Kx + B 限制到S 上的情况(找“n-补”). 逐句评论:
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Let Ks + Bs = (Kx + B)|s.
---- 简化符号。
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The coefficients of Bs belong to Φ(G)...
---- 此句较长,核心 Bs 的系数约束/规律。
---- 涉及两个外部“调用”。
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Since (X, Γ) is plt and -(Kx + Γ)|s ~R (α M - (Kx + Γ))|s is ample, S is Fano type.
---- 粉色部分不就是 A ?
---- 看上去,A|s = α M - (Kx + Γ) = -(Kx + Γ)|s ,但文中的“~R”是怎么回事?
---- 明白了,也许是 M|s ~ 0 (而不是 M|s = 0) 引起了“~R”。
概括: “(X, Γ) is plt” 结合 “-(Kx + Γ)|s is ample” 可推导出 “S is Fano type”。来源待考。
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Thus as -(Ks + Bs) ~ (M - (Kx + B))|s is nef...
---- 参第一段,“M - (Kx + B) ... is still nef...” 看来限制到S后,是保持 nef 的。
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by Theorem 2.13... Ks + Bs has an n-complement Ks + Bs^+ with Bs^+≥ Bs.
---- 看上去,定理 2.13 是决定性的(“n-补”的存在性)。
(回头温习Th2.13)。
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小结:Step3的落点是“Ks + Bs has an n-complement Ks + Bs^+”,对应 “M - (Kx + B) ... is still nef...” 。
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逐句评论是必要的,这个办法能减轻心理压力。
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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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