We investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to “F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic p > 0 p > 0 are characterized by a splitting of the Frobenius map, and define some classes of rings having “mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolution of singularities in characteristic zero. These are defined also for pairs of a normal variety and a Q \mathbb Q -divisor on it, and play important roles in birational algebraic geometry. As an analog of these singularities of pairs, we introduce the concept of “F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair ( A , Δ ) (A,\Delta ) of a normal ring A A of characteristic p > 0 p > 0 and an effective Q \mathbb Q -divisor Δ \Delta on Spec ⁡ A \operatorname {Spec} A . The main theorem of this paper asserts that, if K A + Δ K_{A}+\Delta is Q \mathbb Q -Cartier, then the above three variants of F-singularities of pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analogous to singularities of pairs in characteristic zero.

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