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菲文笔记 | log discrepancy and types of sub-pair

已有 224 次阅读 2021-11-24 23:14 |个人分类:师生园地|系统分类:科研笔记

[注:下文作为笔记的附件,为方便温习及简化笔记的结构。]

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It's going on here for the third round of learning of Birkar's BAB-paper (v2), with scenarios of chess stories. No profession implications. 

Sub-pairs are classified by introducing the concept of "log discrepancy" which is defined as

a(D, X, B): = 1 - [Bw]

where [·] takes the coefficient of Bw on D, a prescribed prime divisor; Bw is the image of B in W resulted from the map called "log resolution"

φ: W --> X

with respect to the sub-pair (X, B). 

Note: [Bw] is a homemade notation for μDBW, the coefficient of Bw on D.

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Three typical types/ states of a sub-pair  ——

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.                 sub-lc,                a ≥ 0

(X, B)   is   sub-klt,         if    a > 0        holds for every D.

                  sub-eps-lc,         a ≥ eps

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It is possible for a log discrepancy to take a non-positive value. Consider a prime divisor D on birational models of X. One call D a non-klt place of (X, B), if a ≤ 0. The image on X of a non-klt place is called non-klt centre. Note that, "non-klt" is replaced by "lc" for lc pairs in some texts, to say lc-place or lc centre [Nov. 21]




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