三次曲线射影分类的研究(一)——对现有分类法的讨论

Study on Projective Classification of Cubic Curves(Ⅰ):A Discussion of Existing Classifications

现有三次曲线的射影分类法有２种：一是牛顿提出的方案。这些成果为进一步研究有关问题提供了重要的基础。对这２种分类法深入分析比较后发现，两者不仅存在着某些差别，而且均有一些不完善之处。其中值得讨论的问题是：１）对于任意２条三阶六级曲线，是否通过一个中心投影将其中一条投影成为另一条；２）对于２条射影结构上具有差别的曲线，是否被归入不同类别之中，本文中提出了否定性的见解。

There are two approaches of projective classifications of the cubic curves in existence:one is Newton's conjecture advanced in about 1676,another is Gordan- Sokolov's classification.These achievements present a useful basis for researching relevant problems.By analyzing and comparing these two approaches,it is discov- ered that there are some differences and some imperfections in each one.Above two problems are deeply discussed:(1)Whether one of the two cubic curves of the 3rd order and 6th class can be project to the other;(2)Whether the two cubic curves having different projective construction can be classified into different classes.The negative answers are given in this paper.