A simple proof for the existence of Zariski decompositions on surfaces
A simple proof for the existence of Zariski decompositions on surfaces
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表面上 Zariski 分解存在性的简单证明
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发表时间：
2007
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通讯作者：
Thomas Bauer
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作者：
Thomas Bauer
分解D=P+N称为D的Zariski分解，因子P和N分别是D的正和负部分。Zariski的结果已被用于研究曲面上的线性级数，并用于曲面的分类(见[1，Chapt.14]和[4，第2.3.E节]，以及其中的提法)。我们还提到，由于Fujita(见[2]和[1]中的NICE帐户)，存在伪有效因子的扩展。给定一个有效因子D，Zariski的原始证明使用了一个相当复杂的过程，从D的那些满足D·C60的分量C中构造出负部分N。这里我们的目的是提供一个快速而简单的证明，基于正部分P可以被构造为D的最大NEF子因子的思想。这个最大值条件在表面上等价于Nakayama的伪有效R-因子的ν分解的定义条件(见下面的注释)。这种方法产生一种计算P的实用算法可能是有用的。
The decomposition D = P + N is called the Zariski decomposition of D, the divisors P and N are respectively the positive and negative parts of D. Zariski’s result has been used to study linear series on surfaces, and in the classification of surfaces (see [1, Chapt. 14] and [4, Sect. 2.3.E], as well as the references therein). We also mention that there is an extension to pseudo-effective divisors due to Fujita (see [2] and the nice account in [1]). Given an effective divisor D, Zariski’s original proof employs a rather sophisticated procedure to construct the negative part N out of those components C of D satisfying D ·C 6 0. Our purpose here is to provide a quick and simple proof, based on the idea that the positive part P can be constructed as a maximal nef subdivisor of D. This maximality condition is in the surface case equivalent to the defining condition of Nakayama’s ν-decomposition of pseudo-effective R-divisors (see the Remark below). It may be useful that this approach yields a practical algorithm for the computation of P .