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In the last period I have studied a number of papers of O. Fujino and F. Ambro using the language of b-divisors. So far it seems to me that every proof I have studied can be translated in the classical language without problems making it, in my opinion, more easily readable. I cannot thus understand what is the strength of this new language. Are there significant improvements it introduces? What are they?

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It can be convenient sometimes, eg: http://arxiv.org/abs/math/0608260. They use this language to define their positive intersection product, which is defined in terms of supremums over all choices of certain nef classes on all birational modifications. Note that they want classes, not numbers, for their statements about the derivative of the volume.

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