Cone theorem for log-canonical fourfolds in positive and mixed characteristic
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Does a cone theorem hold for projective Q-factorial log-canonical fourfold pairs over the eligible positive- and mixed-characteristic bases with p>5? The recorded target assumes log resolutions for the relevant pairs and K_X+Δ≡_U M≥0.
Why it matters and what counts as progress
Why it matters: The fourfold cone theorem is a structural step in the minimal-model program under explicit characteristic and resolution hypotheses.
What counts as progress: Prove the fourfold cone conclusion under these assumptions or remove a stated resolution or effectivity hypothesis.
How the problem developed
- 2015 · Partial result [1]
Christopher Hacon and Chenyang Xu Established major threefold minimal-model-program results in positive characteristic, providing lower-dimensional input for later fourfold work.
- 2023 · Partial result [1]
Christopher Hacon and Jakub Witaszek Proved a relative minimal model program for fourfolds in positive and mixed characteristic, supplying birational tools used by the later cone theorem.
- 14 Aug 2026 · Reported claim [1]
Joe Waldron Proved a cone theorem for effective projective log-canonical Q-factorial fourfold pairs in positive and mixed characteristic p>5, assuming log resolution for birational pairs.
AI contribution
The cone theorem for effective fourfold pairs in characteristic p > 5
Waldron proves a cone theorem for projective Q-factorial log-canonical fourfold pairs over eligible positive- and mixed-characteristic bases p>5, assuming log resolutions for the relevant pairs and numerical equivalence K_X+Δ≡_U M≥0.
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