Length of extremal rays and generalized adjunction
Length of extremal rays and generalized adjunction
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极值射线的长度和广义附加
DOI：
10.1007/bf01215656
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发表时间：
1989
影响因子：
0.8
通讯作者：
J. Wiśniewski
中科院分区：
文献类型：
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作者：
J. Wiśniewski
本文应用Mori理论研究了附加词的一个推广。线性系统IKx+ HI”伴随”到曲面X上的一个充分因子H最初是由Sommese [Sol]和货车de Ven [VdV]用现代术语研究的。关于附加的研究大多集中在奇异曲面和三重曲面上。最近推广的附加函数（[Io]，[So2]，[Fu]）处理极化对（X，H），其中X是n维的集合，H是X上的充分因子。这个问题的一部分是对（X，H）给出一个分类，对于”足够大”的r，因子Kx+ rH不是半充足的（或：不是充足的）。本文考虑光滑簇X上秩为r（“充分大”）的充要张成向量丛E，并研究了伴随因子Kx + clE在什么情况下不是数值有效的.我们得到的结果是：只有那些允许一个极化H使得Kx+ rH不是半满的流形，才有E使得Kx + clE不是有效的.我们的主要工具是Mori理论，通过Mori理论我们理解了关于流形X上的曲线锥的结果和技巧，其中的典型因子Kx在数值上是无效的。虽然考虑曲线的数值性质的最初目的是为了证明Frankel-Hartshorne猜想[Mol]，以及后来的3重分类[Mo 2]，但Mori理论的范围似乎要大得多。特别是，正如它是由几个作者（如[L+ PL，[IO]）注意到，森理论似乎是非常有用的调查adjunction。在第一节中，我们给出了Mori理论和其他相关结果的概述。我们引入极值射线的长度的概念，（1.9），这是在进一步的论证中使用。在第二节中，我们证明了关于具有”足够大”长度的极值射线的流形的一些结果，（2.4），（2.5）。特别地，我们考虑具有切丛的充足第二外幂A2 TX（2.8）的流形的情形。我们证明了在3维空间中，它们只是射影空间和超二次曲面，（2.9）。
In this paper we apply Mori Theory to consider a generalization of adjunction. The linear system IKx+ HI" adjoint" to an ample divisor H on a surface X was originally investigated in modern terms by Sommese [Sol] and Van de Ven [VdV]. Most of the studies concerning adjunction were devoted to singular surfaces and threefolds. Recent generalization of adjunction ([Io],[So2],[Fu]) deals with polarized pairs (X, H), where X is a variety of dimension n and H is an ample divisor on X. A part of the problem is to give a classification of the pairs (X, H) for which the divisor Kx+ rH is not semi-ample (or: not ample), for" sufficiently large" r. In this paper we consider an ample and spanned vector bundle E of (" sufficiently large") rank r on a smooth variety X, and study when the adjoint divisor K x+ clE is not numerically effective. The answer we obtain is as follows: only these manifolds which admit a polarization H such that Kx+ rH is not semiample, can have E such that K x+ clE is not effective. Our main tool is Mori Theory.By Mori Theory we understand the results and techniques concerning the cone of curves on a manifold X whose canonical divisor K x is not numerically effective. Although the original purpose for considering the numerical properties of curves was the proof of Frankel-Hartshorne Conjecture,[Mol], and afterwards, the classification of 3-folds,[Mo2], the scope of Mori Theory seems to be much larger. In particular, as it was noticed by several authors (eg [L+ Pl,[Io]), Mori Theory seems to be very useful in investigating adjunction. In Section 1 we give an outline of Mori Theory and other related results. We introduce the notion of the length of an extremal ray,(1.9), which is used in the further arguments. In Section 2 we prove some results on manifolds having extremal rays of" sufficiently large" length,(2.4),(2.5). In particular, we consider the case of manifolds with ample second exterior power of the tangent bundle, A2TX,(2.8). We prove that in the dimension 3 these are only the projective space and the hyperquadric,(2.9).
SUMIHIRO，Hideyasu：“P^n 上 2 束对称张量的消失定理及其应用”
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