Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties
Frobenius pull-back of vector bundles of rank 2 over non-uniruled varieties
复制标题
2 阶向量束相对于非单规变体的 Frobenius 回拉
DOI：
10.1007/bf01445114
复制
发表时间：
1993
影响因子：
1.4
通讯作者：
A. Moriwaki
中科院分区：
文献类型：
--
作者：
A. Moriwaki
在正特征中，在满射态射下，向量丛的半稳定性不保持。事实上，Gieseker [Gi]在曲线上发现了秩为2的半稳定向量丛，使得它通过绝对Frobenius的拉回是不稳定的。这样的现象使得不可能以原生的方式将Kodaira-Mumford [Reid]的消失定理和Bogomolov-Gieseker不等式从特征零推广到特征p。然而，在不可分态射下的半稳定性的分解可能会揭示关于所讨论的多样性的有趣信息，正如Ekedahl [Ek]和Shepherd-Barron IS-BI]在表面情况下所观察到的那样。本文的目的是用一个简单而普遍的原理来解释他们的结果：如果秩为2的向量丛的Frobenius拉回非常不稳定，那么这个簇是uniruled的（参见图1）。引理1和引理2）。在本文中，我们将固定一个代数闭域k，并且每个代数概型都将定义在k上。我们从两个基本引理开始：引理1（p= char（k）> 0）。设X是d维光滑射影簇，E是X上秩为2的向量丛。设re：IP（E）~ X是E的投射丛，0~（1）是~（E）的重言式线丛.设L是X上的线丛，n>-2为正整数.如果存在IP（E）上的既约且不可约的有效因子Y，使得Y在数值上等价于d）u，CE}（n）--n*（L），则对X上的任何nef Q-线丛H1，9，9 Hd-1，我们有
In positive characteristics, semi-stability for vector bundles is not preserved under surjective morphisms. Indeed, Gieseker [Gi] found a semistable vector bundle of rank 2 over a curve such that its pull-back via the absolute Frobenius is unstable. Such phenomena make it impossible to carry over some beautiful results like the vanishing theorem of Kodaira-Mumford [Reid] and the Bogomolov-Gieseker inequality from characteristic zero to characteristic p in a native way. Nevertheless, the break down of the semi-stability under inseparable morphisms might uncover intriguing information on the variety in question, as was observed by Ekedahl [Ek] and Shepherd-Barron IS-BI] in surface case. The purpose of this note is to explain their results in terms of a simple and general principle: if a Frobenius pull-back of a rank 2 vector bundle is very unstable, then the variety is uniruled (cf. Lemma 1 and Lemma 2). Throughout this note, we will fix an algebraically closed field k and every algebraic scheme will be defined over k. We start with two elementary lemmas.Lemma 1 (p= char (k)> 0). Let X be a smooth projective variety of dimension d and E a vector bundle on X of rank 2. Let re: IP (E)~ X be the projective bundle of E and 0~(~)(1) the tautological line bundle of~(E). Let L be a line bundle on X and n>-2 a positive integer. If there is a reduced and irreducible effective divisor Y on IP (E) such that Y is numerically equivalent to d) u, CE}(n)--n*(L), then, for any nef Q-line bundles H1, 9 9 Hd-1 on X, we have