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菲文笔记 | sub-pair and pair

已有 215 次阅读 2021-11-24 23:04 |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。]

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. 

.

So far, I've not described the formal definition of a pair. Actually, a pair is a special case of the so-called "sub-pair" which is formally defined as follows (See, Sect. 2.2, Birkar 2016b v2.).

sub-pair (X, B) consists of a normal quasi-projective variety X and an lR-divisor B with coefficients in (-oo, 1] such that Kx + B is lR-Cartier.

---- When talking about a sub-pair, three objects are involved.

---- That is, X, B and Kx + B.

---- Each of them has a name and a basic attribution.

---- X is called "variety" with the attribution of "normal quasi-projective", or "nqp" by shorthand.

---- B is called "lR-divisor" with its attribution described by its coefficients which fall into (-oo, 1], in particular.

---- Kx + B is called "operation form" (of the pair) with the attribution of "lR-Cartier".

---- At this stage, the objects as well as their attributions are just taken on the level of names, other than their own definitions.

---- Guess varieties and divisors refer to some kind of "shapes".

---- In a sense, the spirit of algebra is that, one doesn't even have to know what one is playing about.

---- Or, one can safely stay on certain level of names for a long while.

.

So comes the definition of a pair ——

If B ≥ 0, we call B a boundary and call (X, B) a pair.

---- That is, a pair is a sub-pair (X, B) such that the coefficients of B fall into [0, 1], instead of (-oo, 1].  [Nov. 18]

---- Or, a pair takes the form of (X, B) where:

---- X is a normal quasi-projective variety;

---- B is a lR-divisor with coefficients in [0, 1];

---- Kx + B is lR-Cartier. [Nov. 19]




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