Minimal models or Mori fibre spaces for generalized log canonical Q-pairs

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Minimal models or Mori fibre spaces for generalized log canonical Q-pairs

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Take a projective generalized log canonical Q-pair (X, B+M). You can say which endpoint it reaches, a minimal model or a Mori fibre space, and explain why the paper's argument guarantees that endpoint exists.

The statement and how to read it

Everything depends on one divisor, D = K_X + B + M [0:29]. If D is pseudo-effective, a birational map that extracts no divisors reaches a model Y where K_Y + B_Y + M_Y is nef. If D is not pseudo-effective, Y has a contraction h: Y → T with dim T < dim Y, relative Picard number one, and −(K_Y + B_Y + M_Y) ample over T. In both cases the nef b-divisor stays fixed, and no log discrepancy goes down: it strictly increases on divisors that the map contracts. Setting M = 0 gives the ordinary lc case (Corollary 1.2).

Try it on ℙ² with no boundary [1:31]. K = −3L. With M = 2L you get K+M ≡ −L, which is not pseudo-effective, so ℙ² → pt is already a Mori fibre space. With M = 3L you get K+M ≡ 0, which is nef, so ℙ² is its own minimal model. The variety is the same, the nef part changes, and so does the endpoint.

Why it holds: the proof chain

The chain has four links [2:06]: Theorem 3.1 (local phase), then Proposition 3.2 (a smooth canonical MMP with scaling terminates), then the Tsakanikas–Xie NQC theorems, then Proposition 3.4 (descent from ℂ to any algebraically closed field of characteristic zero).

Local phase [2:33]: on Gorenstein klt germs with a cyclic action, normalized-volume minimizers (Blum, Xu, Xu–Zhuang) lead to divisorial valuations with bounded discrepancy. Their values recover ℓ times the character angle, modulo ℤ.
Termination [3:18]: suppose the program is infinite. Then the limits of the non-decreasing, bounded discrepancy changes form a countable set. If the canonical indices were unbounded, index-one covers would put ℓα into that countable set for an arbitrary angle α. That is impossible, so there is a common Cartier index. A divisor would then have to be globally generated and also negative on a contracted curve, which is a contradiction.
Generalized step [4:46]: M cancels when you compare models. The weak Zariski decomposition and Tsakanikas–Xie's theorems give a terminating MMP with scaling on X, and the sign of D picks which endpoint you reach.

What is not claimed

The paper does not say that every MMP terminates, only that one terminating program exists. It needs no Q-factoriality. It does not prove semiampleness for nonzero nef data [5:39]. The rule to take away [5:57]: check the sign of K + B + M first.

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