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.


14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
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