Frobenius non classical curves
Frobenius non classical curves
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弗罗贝尼乌斯非经典曲线
DOI：
10.1007/bf01188523
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发表时间：
1990
影响因子：
0.6
通讯作者：
J. Voloch
中科院分区：
文献类型：
--
作者：
A. Hefez;J. Voloch
1. 介绍。在本文中，我们将关注有限域上定义的投影曲线，其性质是在Frobenius映射下的任何光滑点的像都包含在该点的切线上。我们将得到关于这类曲线的算术和几何的几个结果。这些曲线的研究受到KO St6hr和第二作者[5]的工作的启发，其中对于没有这种和类似病理行为的曲线，可以获得其有理点数量的良好界限。例如，我们证明了这样的曲线是非自反的，并且我们计算了它们在非奇异情况下的有理点的精确数目。对平面曲线的情况作了详细的研究。找到了它们的度的下界和上界，并研究了不等式的极端情况。我们还研究了它们的对偶曲线的几何形状。
1. Introduction. In this paper we will be concerned with projective curves defined over finite fields with the property that the image of any smooth point under the Frobenius map is contained in the tangent line at the point. We shall obtain several results concerning the arithmetic and geometry of such curves. The study of these curves was inspired by the work of KO St6hr and the second author,[5], where for curves without this and similar pathological behaviour, good bounds were obtained for the number of their rational points. We show, for instance, that such curves are non reflexive and we compute the precise number of their rational points in the non singular case. A detailed study of the case of plane curves is done. Lower and upper bounds of their degrees are found and the extremal cases of the inequalities are studied. We also study the geometry of their dual curves.