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The Cone Theorem for Effective Fourfold Pairs in Characteristic $p > 5$

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cone-theorem-effective-fourfold-pairs-char-pAlgebraposed by The char-p minimal model program (Birkar, Hacon, Xu, Waldron and others)recorded: partial

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Statement

Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to XX, the cone theorem holds for projective log canonical, Q\mathbb{Q}-factorial fourfold pairs (X,Δ)(X, \Delta) with KX+Δ≡M≥0K_X + \Delta \equiv M \ge 0, over bases of positive and mixed characteristic p>5p > 5.

Context

The minimal model program in positive characteristic is a central program of modern algebraic geometry, and the cone theorem for fourfolds is a real step it has been waiting for - conditional on log resolution, which keeps it below the unconditional band.

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  • #1

    Attempt 1

    proof attemptChatGPT 5.6 Sol, Codex with Joe Waldron ·
    AI involvement
    ai co developed
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    models
    ChatGPT 5.6 Sol, Codex
    people
    Joe Waldron

    The paper's AI statement, in the author's words: he worked out the outline with all essential ingredients in Spring 2025, but one gap remained that he could not repair; he gave the unfinished proof to ChatGPT 5.6 Sol in Summer 2026 asking it to fill the gap, and it modified the approach to avoid the issue, which after further editing by hand and with Codex became the current version.

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