三次曲线射影分类的研究(二)——射影分类新方案

Study on Projective Classification of Cubic Curves(Ⅱ):A New Plan of Classification

现有三次曲线射影分类存在一些不完善之处，有心要作进一步的研究。为了提出一种分类新方案，对各种三次曲线最简方程和射影结构进行了深入研究。新方案与高尔丹－索科洛夫分类法有２个根本不同点：１）最简方程不同；２）三阶六级曲线被分成３种类型，而原有分类中仅为２种。这一分类法了长期流传的，认为由５条典型曲线通过射影变换即可得到各种不同三次曲线的观点。

The projective classification of cubic curves in existence have some im- perfections judged by the requirements of the projective theory.For finding a new plan it is necessary to study this problem again.In depth,such as finding the most simple equations of cubic curves;distinguishing the projective structures of every cubic curve of various types,and etc.There are two different essential points be- tween the new plan and the Gordan-Sokolov's classification:(1)The most simple e- quations of cubic curves are not alike;(2)The cubic curves of 3rd order and 6th class are classified in three types not only in two as the Sokolov's.Such prevalent viewpoint that arbitrary cubic curve can be obtained from projective transformation of the five canonical curves is negated.