By Edwin H. Spanier

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63 If −1 < a ≤ 0 and jd + a ≤ jd for all j, then d is rational. Proof Assume by contradiction that d is irrational. The cyclic subsemigroup A ⊂ R/Z generated by d is infinite, therefore it is dense. In other words, the fractional parts {jd }, as j is a positive integer, are dense in the open interval (0, 1). For some j, {jd } + a > 0, and then jd + a > jd , a contradiction. 1 The finite generation conjecture implies existence of pl flips In this section, we show that the finite generation conjecture implies the existence of pl flips.

Fix a mobile anti-ample Cartier divisor M on X with support Finite generation on surfaces and existence of 3-fold flips 41 not containing S; write Mi = Mob iM and Di = (1/i)Mi . 36, M• is subadditive and D• is convex. Denote by H 0 (X , iM ) = R(X , D• ) R = R(X , M ) = i the associated pbd-algebra. Denote by R0 = Recall that, by definition, R0i = resS R the restricted algebra. R0i = Im res : H 0 (X , iM ) → k(S) . 40 M•0 is subadditive and D0• is convex). 55 starting with V = R0 . In particular, R0 ⊂ RS is an integral extension and R0 is finitely generated if and only if RS is finitely generated.

When Di = (1/i)Mi , where M• is a subadditive sequence of mobile divisors, we say that M• is the mobile sequence of the pbd-algebra. 50 In the definition, ‘pbd’ stands for pseudo-b-divisorial, as opposed to ‘genuine’ b-divisorial algebras of the form i H 0 X , OX (iD) for a fixed b-divisor D. 51 When X is affine we omit ‘H 0 (X , −)’ from the notation. In the general case, there is a proper birational morphism f : X → Z to an affine variety Z and R(X , D• ) = R(Z, D• ). For some purposes, we can work with Z, D• instead of X , D• , and thus in practice we can assume that X is affine.

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