Let $X$ be a projective Frobenius split variety with a fixed Frobenius splitting $\theta$. In this paper we give a sharp uniform bound on the number of subvarieties of $X$ which are compatibly Frobenius split with $\theta$. 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.

Let be a proper ideal of a commutative Noetherian ring R of prime characteristic p and let Q() be the smallest positive integer m such that ${{(}^F)}^{\left[p^m\right]}{=}^{\left[p^m\right]}$, where ${}^F$ is the Frobenius closure of . This paper is concerned with the question whether the set $Q{(}^{\left[p^m\right]}):m\in \mathbb{N}₀$ is bounded. We give an affirmative answer in the case that the ideal is generated by an u.s.d-sequence c₁,..., cₙ for R such that (i) the map $R/{\sum }_{j=1}^{n}R{c}_j\to R/{\sum }_{j=1}^{n}Rc{²}_j$ induced by multiplication by c₁...cₙ is an R-monomorphism; (ii) for all $\in ass(c{₁}^j,...,c{ₙ}^j)$, c₁/1,..., cₙ/1 is a $R_{}$-filter regular sequence...