A Finiteness Theorem for Del Pezzo Surfaces over Algebraic Number Fields
A Finiteness Theorem for Del Pezzo Surfaces over Algebraic Number Fields
复制标题
代数数域上Del Pezzo曲面的有限定理
DOI：
10.1112/jlms/s2-32.1.31
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发表时间：
1985
影响因子：
1.2
通讯作者：
A. Scholl
中科院分区：
文献类型：
--
作者：
A. Scholl
Faltings [5]最近证明了，给定一个代数数域K，一个K的素数的有限集合P，和一个正整数g，K上g维的阿贝尔簇的同构类的个数是有限的，这些同构类具有良好的约化。一个直接的结果是相应的曲线的有限性定理（Shafarevich猜想[13]）。很自然地会问类似的有限性定理是否对其他类型的代数簇成立。在这篇文章中，我们处理的情况下，del Pezzo曲面。一般来说，主要问题之一是给“好约化”下一个恰当的定义。对于del Pezzo曲面，它是由反正则级数自动极化的，这不会引起任何困难，但见下面的注释4.6，关于“明显”定义的失败。
Faltings [5] has recently proved that, given an algebraic number field K, a finite set P of primes of K, and a positive integer g, the number of isomorphism classes of abelian varieties of dimension g over K with good reduction away from P is finite. An immediate consequence is a corresponding finiteness theorem for curves (the Shafarevich conjecture [13]).It is natural to ask whether similar finiteness theorems hold for other classes of algebraic varieties. In this article we treat the case of del Pezzo surfaces. In general, one of the principal problems is to give an appropriate definition of'good reduction'. For del Pezzo surfaces, which are automatically polarised by the anticanonical series, this causes no difficulty, but see Remark 4.6 below for the failure of an'obvious' definition.