Log-canonical threefold abundance in numerical dimension one

What OpenAI claims

Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.

Papers (2)

1. Log abundance in numerical dimension one for threefolds in positive characteristic

   5 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If (X,B) is a projective log canonical threefold pair with effective rational boundary, and K_X+B is ℚ-Cartier, nef, and of numerical dimension one, then K_X+B is semiample. Neither terminality nor ℚ-factoriality of X is required.

2. Abundance in numerical dimension one for terminal threefolds in positive characteristic

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if KX is nef with ν(K_X)=1, then KX is semiample and κ(X,K_X)=1.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.