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Corollary 8 ([Horizontal variants]). The chart and metric conclusions have the following two variants. The same tailorable bounded complement and homogeneous chart argument works if we are working over an algebraically closed generic …

Lemma 7 ([Degrees control angular variation]). On a chart of Lemma 6 ([Bounded toroidal charts]), the order of a nonzero rational function is continuous, homogeneous and finite piecewise linear, including on the faces. If its pole divisor …

Proposition 5 ([Uniform charts]). At a chosen vertical place in Lemma 4 ([Tools at an lc place])(ii), after a bounded degree extension of k(W) and a chosen extension of the DVR and of the place, there is a toroidal cone representing the …

Lemma 4 ([Tools at an lc place]). Let g:X1→W be a Mori step as above, and let A* be an effective generalized CY discrepancy on X1 over S . Suppose that A* is lc over the generic point of W , and that an effective generalized klt CY …

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For (i), tailor the complement on the Fano type model using the klt perturbation in Lemma 4 ([Tools at an lc place]), and then push the complement and the effective CY data crepantly to the bounded normal small model. Their total divisors are -linearly trivial. This supplies the bounded embedded marked family and all the degree estimates in Proposition 5 ([Uniform charts]); the anticanonical degree is bounded by the same curve slicing argument. For (ii), use the two-point construction for each of the two b-nef parts, taking the pushed ample errors to be o(ξ) . As above replace those parts by general effective representatives after the same arbitrarily tiny ample addition on a high determination, avoiding both centres. The residual rational principal term for N now uses functions with pole degree divided by denominator O(ξ) on F ; apply Lemma 7 ([Degrees control angular variation]) (using constancy if pole degree <1 ).

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