Minimal Models of Canonical 3-Folds
Minimal Models of Canonical 3-Folds
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DOI：
10.2969/aspm/00110131
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发表时间：
1983
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通讯作者：
M. Reid
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文献类型：
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作者：
M. Reid
§ - oo. 摘要 本文引入了三维簇的极小模型的一个临时定义（0.7），并在额外假设下对其进行研究。主要结果是定理（0.6），其中我刻画了必然出现在极小模型上的奇点，并通过对[C3 - f]中所研究的典范模型X进行爆破，紧密模仿杜瓦尔曲面奇点的极小消解，证明了有限生成一般型三维簇的极小模型S的存在性。除了[C3 - f]中熟悉的技术（微分赋值的计算；循环覆盖；非cDV的指标I点的恰当爆破），主要的新要素（定理（2.6））是一种爆破一维奇异轨迹的方法，它基于布里斯克恩 - 秋里娜关于一族杜瓦尔曲面奇点同时消解存在性的结果，以及伯恩斯和拉波波特在（2）-曲线中的初等变换。第二部分致力于阐述这些初等变换；其中大部分是民间流传的内容，但详细阐述似乎是高维双有理几何的一个关键现象似乎是值得的。[C3 - f]中以及此处引入的典范奇点和终端奇点具有很强的归纳性质，并且有一些理由相信终端奇点将为森重文结果的归纳推广提供自然的范畴：由森锥的K < 0部分的极面所确定的初等收缩（当它们存在时）总是非等奇的。我在§4中包含了关于三年或四年后三维簇的极小模型理论和分类将会是什么样的猜想，以及在§8中的一节猜想，试图确定非直纹情形下极小模型的非唯一性。
§-oo. Abstract This paper introduces a temporary definition of minimal models of 3-folds (0.7), and studies these under extra hypotheses. The main result is Theorem (0.6), in which I characterise the singularities which necessarily appear on a minimal model, and prove the existence of a minimal model S of a 3-fold of f.g. general type, by blowing up the canonical model X studied in [C3-f], imitating closely the minimal resolution of Du Val surface singularities. Apart from techniques familiar from [C3-f] (computations of the valuations of differentials; cyclic covers; crepant blow-ups of index I points which are not cDV), the main new element (Theorem (2.6)) is a method of blowing up the I-dimensional singular locus, based on the Brieskorn-Tyurina result on the existence of simultaneous resolutions of a family of Du Val surface singularities, together with the elementary transformations in ( 2)-curves of Burns and Rapoport. Part II is devoted to an exposition of these elementary transformations; much of this is folklore material, but it seems worthwhile to give a detailed account of what seems to be a key phenomenon of higher-dimensional birational geometry. The canonical and terminal singularities introduced in [C3-f] and here have strong inductive properties, and there is some reason for believing that terminal singularities will provide the natural category for an inductive extension of Mori's results: elementary contractions (when these exist) specified by extremal faces of the K<O part of the Mori cone are always discrepant. I have included in § 4 conjectures as to what the theory of minimal models and classification of 3-folds will look like in 3 or 4 years' time, and a section of conjectures in § 8 attempting to pin down the non-uniqueness of minimal models in the non-ruled case.