This is coming to you from Yiwei LI (PhD, Applied math), Taiyuan University of Science and Technology  (TYUST) Taiyuan, China

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. 

Presentation of a work, generally called "paper", is just a projection of the "camera", often losing information in one way or another.

---- "camera" is a collection of the key elements as well as the contained manipulations and contexts.

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Th 2.15    Th 1.8            ♖    ♘

     ↓   ↖     ↓

Th 1.1      Th 1.6            ♔    ♗

    Mathematics vs Palace stories.(v2)

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Note: technical theorem is not on the board.

♙ ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁺⁻⁰ 1

(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 check... 

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Summary comment: This first part of Para two (of Step 7) is to pave the way, showing the "thought path".

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↑↓ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ α Δ δ μ ≠ ⌊ ⌋ ⌈ ⌉ ∨∧∞Φ⁺⁻⁰ 1

Calling graph for the technical theorem (Th1.9) ——

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Th1.9

    |

[5, 2.13(7)]   Lem 2.26   Pro4.1   Lem2.7

                                                          |

.....................................................Lem2.3   

Note: Th1.9 is only called by Pro.5.11, one of the two devices for Th1.8, the executing theorem.

Pro4.1                                                    

    |

[5, ?]   [37, Pro3.8]   [5, Lem3.3]   Th2.13[5, Th1.7]   [16, Pro2.1.2]  [20]  [25, Th17.4]

Completed notes of the first round learning for v2 Pro.4.1 are packaged on RG.

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Special note: Original synthesized scenarios in Chinese for the whole proof of v1 Th1.7, the technical theorem.

*It's now largely revised* due to new understandings.

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See also: Earlier comments in Chinese* (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.