相容Frobenius分裂子簇、素$F$-理想和对数标准中心的个数
On the number of compatibly Frobenius split subvarieties, prime $F$-ideals, and log canonical centers
摘要：
设$X$是代数闭域上的Frobenius分裂簇，其上有分裂的$θ: F_* Ø_X \to Ø_X$.本文给出了$θ$相容分裂的$X$的子簇个数的一个精确界.特别地，假设$\sL$是$X$上具有$n = \dim H^0(X, \sL)$的足够充足的线丛（例如，如果$\sL$诱导投射正常嵌入）。证明了$X$的$d$维不可约子簇被$θ$相容分裂的个数小于或等于${n \choose d+1}$.这推广了一个众所周知的结果，闭点的数目兼容分裂由一个固定的分裂$θ$。类似地，我们给出了$F$-有限$F$-纯局部环的素$F$-理想个数的一个界.可裂子簇与对数典范中心密切相关。我们的方法适用于任何特征，所以我们也能够绑定通过一个封闭点$x \in X$的日志规范对$(X, Δ)$的日志规范中心的可能数量。具体地，如果$n$是$X$在$x$处的嵌入维数，则$(X, Δ)$至$x$的$d$维对数正则中心的数目小于或等于${n \choose d}$。
英文摘要：
Let $X$ be a projective Frobenius split variety over an algebraically closed field with splitting $θ: F_* Ø_X \to Ø_X$. In this paper we give a sharp bound on the number of subvarieties of $X$ compatibly split by $θ$. 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 $θ$ 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 $θ$. 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 $(X, Δ)$ 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 $(X, Δ)$ through $x$ is less than or equal to ${n \choose d}$.
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