Unstable vector bundles and linear systems on surfaces in characteristicp
Unstable vector bundles and linear systems on surfaces in characteristicp
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DOI：
10.1007/bf01243912
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发表时间：
1991-12
影响因子：
3.1
通讯作者：
N. Shepherd-barron
中科院分区：
文献类型：
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作者：
N. Shepherd-barron
Bogomolov证明了在特征值为0的曲面X上，任意2阶局部自由轴c2 > 4c2都是不稳定的（在某种意义上将在下面回顾），并推广了Castelnuovo和de Franchis的引理，证明了I2}没有正可逆子轴。由此他推导出X的Chern数满足不等式c2 < 4c2（该界后来被Miyaoka[5]和Yau 1-9]降低为c2 < 3c2）。Mumford展示了1-6]如何使用Bogomolov的结果给出Kodaira消失定理的另一个证明，Reider[7]用它改进了Bombieri关于一般类型曲面的多范式映射的结果[2]。在特征p中，Bogomolov的结论不再成立。然而，本文的目的是要表明，这些定理的失败具有显著的后果；例如，我们可以恢复Ekedahl关于Kodaira消失定理的结果[4,11]。1.6和II。[1.7]，以及他的论文中最多隐含的一些改进，并显著改进了他关于多canonical映射的结果1-4，p. 97，主要定理]。
Bogomolov showed [1] that on a surface X in characteristic zero, any rank 2 locally free sheaf with c 2> 4c2 is unstable (in a sense recalled below), and also generalized a lemma of Castelnuovo and de Franchis to show that I2} has no positive invertible subsheaf. From this he deduced that the Chern numbers of X satisfy the inequality c 2< 4c2 (a bound subsequently lowered by Miyaoka [5] and Yau 1-9] to c 2< 3c2). Mumford showed 1-6] how to use Bogomolov's result to give another proof of the Kodaira vanishing theorem and Reider [7] has used it to improve Bombieri's results [2] on pluricanonical mappings of surfaces of general type.In characteristic p Bogomolov's results as stated no longer hold. The aim of this paper is to show, however, that the failure of these theorems has significant consequences; for example, we can recover Ekedahl's results on the Kodaira vanishing theorem [4, I1. 1.6 and II. 1.7], together with some improvements that are at most implicit in his paper, and to improve markedly his results on pluricanonical mappings 1-4, p. 97, Main Theorem].