F-regular and F-pure rings vs. log terminal and log canonical singularities
F-regular and F-pure rings vs. log terminal and log canonical singularities
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DOI：
10.1090/s1056-3911-01-00306-x
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发表时间：
2000-02
影响因子：
1.8
通讯作者：
Nobuo Hara;Kei-ichi Watanabe
中科院分区：
文献类型：
--
作者：
Nobuo Hara;Kei-ichi Watanabe
在本文中，我们研究了 F 正则环（分别为 F 纯环）和对数终端奇点（分别为对数规范）奇点的关系。此外，我们将 F-正则性和 F-纯度的概念扩展到“F-奇点对”。特征 $p>0$ 中的 F-正则环和 F-纯环的概念以 Frobenius 映射的分裂为特征，并定义了一些具有“温和”奇点的环类。另一方面，存在通过特征零奇点的解析定义的对数终端和对数正则奇点的概念。这些也被定义为一对正规簇和其上的 $\Bbb Q$-除数 $\Delta$，并且在双有理代数几何中发挥着重要作用。作为特征零中这些对奇点的类比，我们定义了“对的 F-奇异性”的概念，即特征 $p > 0$ 的正则环 $A$ 的对 $(A,\Delta)$ 的强 F-正则性、除数 F-正则性和 F-纯度，以及 $Y = Spec A$ 上的有效 $\Bbb Q$-除数 $\Delta$。本文的主要定理断言，如果 $K_Y + \Delta$ 是 $\Bbb Q$-Cartier，则上述 F-奇点对的三个变体分别意味着 KLT、PLT 和 LC 性质。我们还证明了对的 F 奇点的一些结果，这些结果类似于特征零中对的奇点。
In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\Bbb Q$-divisor $\Delta$ on it, and play an important role in birational algebraic geometry. As an analog of these singularities of pairs in characteristic zero, we define the notions of "F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta)$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\Bbb Q$-divisor $\Delta$ on $Y = Spec A$. The main theorem of this paper asserts that, if $K_Y + \Delta$ is $\Bbb Q$-Cartier, then the above three variants of F-singularitiesof pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analoguous to singularities of pairs in characteristic zero.