Finite Generation for klt Generalized Pairs
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Statement
The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural to ask whether finite generation survives. It does not: Liu and Wang construct a smooth projective klt generalized pair whose generalized log canonical ring is not a finitely generated algebra.
Context
The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and welcome corrections.
Asks whether the BCHM finite generation theorem, one of the landmark results of modern birational geometry, extends to generalized pairs. Only three years old, which caps it, but leading figures in the field had tried it without success.
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Stated in the abstract, which is unusual: "The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol-ultra, Fable 5, and the Danus system." Remark 1.5 adds that Danus is a specialised agent built on the Rethlas system, and that human verification and polishing were done afterwards.
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