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On the number of compatibly Frobenius split subvarieties, prime $F$ -ideals, and log canonical centers

Karl Schwede; Kevin Tucker — 2010

Annales de l’institut Fourier

Let $X$ be a projective Frobenius split variety with a fixed Frobenius splitting $θ$ . In this paper we give a sharp uniform bound on the number of subvarieties of $X$ which are compatibly Frobenius split with $θ$ . Similarly, we give a bound on the number of prime $F$ -ideals of an $F$ -finite $F$ -pure local ring. Finally, we also give a bound on the number of log canonical centers of a log canonical pair. This final variant extends a special case of a result of Helmke.

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