发表机构
New Mexico State University（新墨西哥州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文构造并研究了实平面四点构型在相似变换群的Nash子群作用下的Nash流形，通过引入四边形概念证明轨道空间为Nash流形，并定义几何不变量aspect，将非例外模问题简化为两个模型纤维的分析。
AI 中文摘要
我们构造并研究了实平面中四点构型在相似变换群的Nash子群作用下的Nash流形。为此，我们使用四边形这一更精细的概念来代替四点构型，因为四边形在退化过程中保留了极限线数据。我们证明了四边形空间${\mathcal Q}$是一个8维Nash流形，并且对于相似群的每个Nash子群$G$，轨道空间${\mathcal Q}/G$都是Nash流形。我们定义了一个称为“aspect”的几何不变量，其取值在$[-1,1]$中，并证明了在区间$(-1,0)$和$(0,1)$上，${\mathcal Q}/G$的相应部分与区间和固定纤维的乘积Nash微分同胚。因此，模问题中的非例外部分归结为对两个模型纤维的分析，每个纤维在四边形本身方面都有自然的几何解释。
英文摘要
We construct and study the Nash manifold of four-point configurations in the real plane modulo the action of a Nash subgroup of the group of similarity transformations. We do so by using the finer notion of a quadrangle in place of that of a four-point configuration, since this retains limiting line data in degenerations. We prove that the space ${\mathcal Q}$ of quadrangles is an 8-dimensional Nash manifold and that, for every Nash subgroup $G$ of the similarity group, the orbit space ${\mathcal Q}/G$ is a Nash manifold. We define a geometric invariant, called aspect, with values in $[-1,1]$, and prove that, over each of the intervals $(-1,0)$ and $(0,1)$, the corresponding part of ${\mathcal Q}/G$ is Nash diffeomorphic to the product of the interval with a fixed fiber. Thus the nonexceptional part of the moduli problem reduces to the analysis of two model fibers, each having a natural geometric interpretation in terms of the quadrangles themselves.
Comments34 pages, 5 figures. Comments welcome!
