2009/05/31 by Karl Schwede, Karl E. Schwede, Karen E. Smith · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algorithm #Combinatorics #Cone (formal languages) #Converse #Discrete mathematics #Divisor (algebraic geometry) #Fano plane #Geometry #Gravitational singularity #Mathematical analysis #Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation #Prime (order theory) #Pure mathematics #Type (biology) #Variety (cybernetics) #Zero (linguistics) #math.AC #math.AG #msc:13A35 #msc:14B05 #msc:14J45
paper · pdf · doi:10.1016/j.aim.2009.12.020
published as Advances in Mathematics, Vol 224 Issue 3, 863--894, 2010 · 31 pages, minor changes throughout. The presentation of section 5 improved. To appear in Advances in Mathematics
arxiv created 2009/12/23 · openalex publication_date 2010/01/22 · arxiv updated 2010/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We prove that every globally F-regular variety is log Fano. In other words, if a prime characteristic variety X is globally F-regular, then it admits an effective \bQ-divisor Δ such that -KX - Δ is ample and (X, Δ) has controlled (Kawamata log terminal, in fact globally F-regular) singularities. A weak form of this result can be viewed as a prime characteristic analog of de Fernex and Hacon's new point of view on Kawamata log terminal singularities in the non-\bQ-Gorenstein case. We also prove a converse statement in characteristic zero: every log Fano variety has globally F-regular type. Our techniques apply also to F-split varieties, which we show to satisfy a "log Calabi-Yau" condition. We also prove a Kawamata-Viehweg vanishing theorem for globally F-regular pairs.