[注：下文是 2 月 25 日发出的群邮件内容，标题为原有的。]

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. 

Acquaintance, understanding and order form the gravity in the world of knowledge.

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

Last learning note was formed on Feb. 2, three weeks ago. 17 pieces of learning notes have been collected and packaged*, for v2 Pro.4.1 of Th1.9, the technical theorem. I expect to give an experimental lecture in my class (undergraduate). This time is to start over for a better acquaintance, with a bit of formality ——

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I decide the statement of Th 1.9 takes a shape of hexagon, arranging the six items at the vertices. An auxiliary graph is needed to indicate the path of sight. Geometry is a study of shapes, say, triangles, rectangles, and circles, etc.. If one puts a big circle and a small circle together, a new shape is formed, say oO. This is a kind of geometry in an algebraic way: o + O = oO. As algebra resorts to symbols to denote any object of concern, one can assign symbols to geometric objects, i.e. shapes. After that, one can focus on these symbols, as well as the operations among them. One more thing, objects may have attributions, just like the situation one encounters in the object-oriented programming (OOP).

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The statement of Theorem 1.9 involves two natural numbers d and p on the input, to produce another natural number n on the output. Among the constraints, one encounters a "pair". This is always the case thorough the whole paper under learning. One can imagine the pair as a symbol of royal court: there is a King and there is a Bishop, borrowing the terms from chess. It looks like (♔, ♗). A royal court may have a flag, to indicate its attribution or state. All together, it is presented as (X, B) with a state called "projective lc", or "plc" for short, like a flag with some strange logo. Both X and B can be viewed as certain geometric objects, or simply shapes. X has the strange name of "variety" while B is called "boundary". At the present stage, the origin or meaning behind these names are not concerned. X has a dimension of d. This appears the only the meaning of the natural number d. The number p is used as a scaling factor for B, such that pB has integral coefficients. In this chess story, there is a role of Noble, illustrated as ♖. This Noble, also denoted by M, has a strange title of "semi-ample Cartier divisor". This title endows M the ability to define a map called "contraction", i.e. f: X --> Z. The capital letter Z denotes a magic zone over which X gains a title of "Fano type". One will see names and titles are keys to open the relevant doors. So far, it is introduced "pair" or royal court, its key members King and Bishop, plus an independent Noble, together with their state(s), names or titles. The net entities are (X, B) and M. The two bodies have a key interaction. To this end, it is convenient to introduce the "operation form" of (X, B), that is Kx + B. The key interaction just takes the form of M - (Kx + B), with a name of "confrontation", also denoted as c(M) for short, owning a strange title of "nef and big" (or just "nb" for short). Lastly, the pair (X, B) has a special appendage called "non-klt centre", denoted by S. So comes a collateral interaction between M and S of (X, B): M|S ≡ 0. This ends of the description of the six items of the statement of Theorem 1.9, summarized in a hexagon graph ——

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                        M

        c(M)nb                M|S ≡ 0 

                      d,p

          XF                     pB

                       plc

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Illustration for the six items of Th1.9.

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                         3

          5                           6

                         0

          4                           2

                         1

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    Path of sight for the illustration.

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So comes the description of the output part of the statement of Theorem 1.9. Together with the natural number n, one obtains another pair (X, Λ) which has a state of lc over a neighbourhood of z: = f(S). Further more, this new pair is also constrained in the term of operation form: n(Kx + Λ) ~ (n + 2)M. Here, the specific integral number in n + 2 is called "shift". This constraint on the operation form can be viewed as a measurement for the new pair: up to a shift of 2, the pair is equivalent to M in its operation form. To apply the concept of "confrontation", this constraint can be expressed as -n/2·c(M)Λ ~ M.

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Problem: What is the state of c(M)Λ ? (semi-ample, ample or nef and big ?).

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

Th1.9

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[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]

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).

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