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菲文笔记 | Technical theorem (v2) ---- camera and projection

已有 308 次阅读 2021-8-14 07:51 |个人分类:师生园地|系统分类:科研笔记

This is coming to you from continued) camera ={R, S, ...}.

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Step 7, Para two (a) ——

Let Λ := B⁺ := B + R.

---- The core task is to construct the increment R (of B), such that (X, B + R) keep lc near S.

---- Guess this notation of Λ is just out of the reviewer's suggestion.

---- The origin of R has been treated in the last note.

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New comments: It appears not proper to say the "origin" of R.

---- Just that, the intention of this theorem is to construct B⁺, or more essentially R, in the context of relative n-complement.

---- I decide R is an element of the "camera".

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For consistency of notation, for the rest of the proof we will use B⁺ instead of Λ.

---- B⁺ appears more intuitive, hinting the connection to B [on] the level of notation.

---- Λ appears to hide the connection, so that the statement of the theorem is more [encapsulated].

---- Also, one feels less abrupt [by] [avoiding] seeing the complicated structure of G too earlier.

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New comments: Theorem 1.9 is a proposition other than a theorem.

---- Lemma is of light statement and light proof.

---- Therorem is of light statement and heavy proof.

---- Proposition is of large statement and heavy proof.

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By construction, n(Kx + B⁺) ~ (n + 2)M.

---- This meets the second item on the outputs of the theorem.

---- Deeper consideration remains to be explored.(?)

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New comments: The derivation of n(Kx + B⁺) ~ (n + 2)M has been shown in the last note, related to the account for nR:= G.

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Special comments: There is something strange here ——

---- If one views Kx + B⁺ as the n-complement of Kx + B, one has n(Kx + B ⁺ ) ~ 0 by the definition given in the paper.

---- Should one expect (n + 2)M ~ 0 ? It appeas not.

---- Here is the position to retype the annotation lines under the statement of Th1.9 (v2 p.6) ——

Note that Kx + Λ is actually a relative n-complement of Kx + B over a neighbourhood of z in the sense of [5, 2.18].

---- So, the concept of "n-complement" here is only relatively understood, like a variant.

The important point here is that the complement is not an arbitrary one since Λ is somehow controlled globally by M as it satisfies the formula n(Kx + Λ) ~ (n + 2)M.

---- This is a further explanation that "the complement is not an arbitrary one", globally controlled by M.

---- One will see in Th1.8, M has a positive lower bound.

---- Repeat: deeper consideration remains to be explored.

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New comments: Just that, one cannot see the usage of n(Kx + Λ) ~ (n + 2)M at present.

---- I expect its usage will be shown in the calling context, i.e. Pro. 5.11.

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It remains to show that (X, B⁺ ) is lc over z = f(S).

---- This is to meet the primary item on the outputs of the theorem.

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First we show that (X, B⁺ ) is lc near S: this follows from inversion of adjunction [20], if we show Ks + Bs ⁺ = (Kx + B ⁺ )|s which is equivalent to showing R|s = Rs.

---- The key is in [20], a paper by M. Kawakita; Inversion of adjunction on log canonicity, Invent. Math. 167 (2007), 129-133.

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New comment: Kawakita (2007) is only five pages, worthy of a * (v1).

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It is my hope that this action would not be viewed from the usual perspective that many adults tend to hold.




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