The Geometric Bogomolov Conjecture for Curves of Small Genus
The Geometric Bogomolov Conjecture for Curves of Small Genus
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DOI：
10.1080/10586458.2009.10129049
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发表时间：
2009-01
影响因子：
0.5
通讯作者：
Xander Faber
中科院分区：
文献类型：
--
作者：
Xander Faber
Bogomolov猜想是关于定义在整体域上的光滑完备曲线上的小高度代数点的有限陈述。我们证明了特征为零的函数域上亏格至多为4的所有曲线的Bogomolov猜想的一个有效形式。我们恢复了亏格2曲线的已知结果，并在许多情况下改进了亏格3曲线的已知上界。对于许多归约不良的亏格为4的曲线，这一猜想以前是不成立的。
The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recover the known result for genus-2 curves and in many cases improve upon the known bound for genus-3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.