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Globally \(F\)-regular and log Fano varieties. (English) Zbl 1193.13004

Globally \(F\)-regular varieties were introduced by K. E. Smith [Mich. Math. J. 48, Spec. Vol., 553–572 (2000; Zbl 0994.14012)] where it was shown that Fano varieties in characteristic \(0\) reduce to globally \(F\)-regular varieties in characteristic \(p\gg 0\). It is known that globally \(F\)-regular varieties satisfy Kodaira type vanishing results and are locally \(F\)-regular (in particular they are Cohen-Macaulay and have pseudo-rational singularities). Note that locally \(F\)-regular singularities correspond to Kawamata log terminal singularities in the context of the minimal model program.
The main result of the paper under review is to show that if \(X\) is a globally \(F\)- regular variety of characteristic \(p\neq 0\), then there exists an effective \(\mathbb Q\)-divisor \(\Delta\) such that \((X,\Delta )\) is a log Fano variety (i.e. \(-(K_X+\Delta)\) is ample and \((X,\Delta )\) is Kawamata log terminal). The authors also show that log Fano varieties in characteristic \(0\) reduce to globally \(F\)-regular varieties in characteristic \(p\gg 0\).

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14J45 Fano varieties
14B05 Singularities in algebraic geometry

Citations:

Zbl 0994.14012
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References:

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