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Surfaces of globally $$F$$-regular type are of Fano type. (English) Zbl 1374.14038
Let $$X$$ be a normal projective variety, and $$\Delta$$ an effective $$\mathbb Q$$-divisor on $$X$$, such that $$K_X +\Delta$$ is $$\mathbb Q$$-Cartier. If the singularity of the pair $$(X,\Delta)$$ are at worst klt and $$-(K_X+\Delta)$$ is ample, then the pair is called log Fano. A normal projective variety $$X$$ is called of Fano type if there exists $$\Delta$$ as above such that $$(X,\Delta)$$ is log Fano.
Global $$F$$-regularity is a notion defined using the Frobenius morphism in characteristic $$p$$ which makes sense in characteristic zero by reduction to characteristic $$p$$ (see [K. E. Smith, Mich. Math. J. 48, 553–572 (2000; Zbl 0994.14012)]).
In the paper under review, a conjecture of K. Schwede and K. E. Smith [Adv. Math. 224, No. 3, 863–894 (2010; Zbl 1193.13004)] claiming the equivalence of being a variety of Fano type and being globally $$F$$-regular is investigated and proven in the case of surfaces.

MSC:
 14J45 Fano varieties 14E30 Minimal model program (Mori theory, extremal rays)
Citations:
Zbl 0994.14012; Zbl 1193.13004
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