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

Abstract

Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $\theta : F_* \O_X \to \O_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $\theta$. In particular, suppose $\sL$ is a sufficiently ample line bundle on $X$ (for example, if $\sL$ induces a projectively normal embedding) with $n = \dim H^0(X, \sL)$. We show that the number of $d$-dimensional irreducible subvarieties of $X$ that are compatibly split by $\theta$ is less than or equal to ${n \choose d+1}$. This generalizes a well known result on the number of closed points compatibly split by a fixed splitting $\theta$. Similarly, we give a bound on the number of prime $F$-ideals of an $F$-finite $F$-pure local ring. Compatibly split subvarieties are closely related to log canonical centers. Our methods apply in any characteristic, and so we are also able to bound the possible number of log canonical centers of a log canonical pair $(X, \Delta)$ passing through a closed point $x \in X$. Specifically, if $n$ is the embedding dimension of $X$ at $x$, then the number of $d$-dimensional log canonical centers of $(X, \Delta)$ through $x$ is less than or equal to ${n \choose d}$.