Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of compatibly Frobenius split subvarieties, prime
  $F$-ideals, and log canonical centers   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 ${n \choose d}$.

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The categories of this paper are: funct-an
33
The categories of this paper are: plasm-ph
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The categories of this paper are: stat
5
The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph