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This is an in-mail from TYUST.
本期开始加开窗口,推出科学网特色博主,有用链接等。
...他解释说他的大部分工作都标记着“服务的态度”*。
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(接上回*)Lemma 3.2. Assume that Theorem 1.6 holds in dimension ≤ d and that Theorem 1.1 holds in dimension ≤ d - 1. Then Theorem 1.4 holds in dimension d.
评论:此引理的证明有两大段 “Pick...Since...”,上回读写了第一段,此次读写第二段。
Since (Y, BY + s LY) is klt weak log Fano, Y is Fano type.
既然 (Y, BY + s LY) 是 klt weak log Fano, 则 Y 是 Fano type.
注:前半句突然,像是“显然”,但不晓得怎么看出来的(?)。整句是通的。
Run an MMP on -T and let Y' --> Z' be the resulting Mori fibre space. Then
- (KY' + BY' + sLY') ~ R(1-s)LY' ≥ 0.
在 -T上运行MMP,并令 Y' --> Z' 为 Mori 纤维空间。则有
- (KY' + BY' + sLY') ~ R (1-s) LY' ≥ 0.
注:T和Y是上一段产生的(见 Pick...),Y' 和 Z' 是这儿新产生的。这里第二句看不出(?)。
Moreover, (Y', BY' + s LY') is eps'-lc because (Y, BY + s LY) is eps'-lc and -(KY + BY + s LY) is semi-ample.
进一步,(Y', BY' + s LY') 是 eps'-lc的,因为 (Y, BY + s LY) 是 eps'-lc的并且 -(KY + BY + s LY) 是 semi-ample的.
注:此段第一句说“(Y, BY + s LY) is klt weak log Fano”,而此处又说“ (Y, BY + s LY) is eps'-lc”,似乎用到eps'-lc的双重性(凡是eps-lc,都是klt?) 。
If dim Z'>0, then restricting to a general fibre of Y' --> Z' and applying Theorem 1.4 in lower dimension by induction (or applying Theorem 1.1) shows that the coefficients of the horizontal/Z' components of (1 - s)LY' are bounded from above.
若 dim Z'>0,则限制到 Y' --> Z' 的一般的纤维,并通过归纳法应用低维的定理1.4(或应用定理1.1),表明 (1-s)LY' 的横向/Z' 分量的系数存在上界。
注:没感觉了(?)。
In particular, μT'(1-s)LY' is bounded from above. Thus from the inequality
μT'(1-s)LY' ≥ (1-s)(eps-eps')/s,
we deduce that s is bounded from below away from zero.
特别地, μT'(1-s)LY' 存在上界。则由不等式
μT'(1-s)LY' ≥ (1-s)(eps-eps')/s,
可推出 s 存在正的下界.
注:(1-s)两边约掉了(?)。
Therefore, we can assume Z' is a point and that Y' is a Fano variety with Picard number one. Now
- KY' ~ R(1-s)LY' + BY' + sLY' >= (1-s)LY',
so by Proposition 3.1, muT'(1-s)LY' is bounded from above which again gives a lower bound for s as before.
因此,可假定Z'是一个点而Y'是个带有Picard数1的Fano型簇。现在
- KY' ~ R(1-s)LY' + BY' + sLY' >= (1-s)LY',
因此由命题3.1,muT'(1-s)LY'存在上界,这再次给出s的下界(如之前那样)。
注:没感觉。
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小结:完成引理3.2第二段证明的读写(效果上,只能说是“泡”了一会儿数学~ )。
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临时想法:也许,读写定理的证明时,也该从后往前走!(更加彻底的“回到顶端”,其实就是彻底的逆向法、倒着走...)。
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