Pascal's theorem gives a synthetic geometric condition for six points $a,\dots ,f$ in ${\mathbb{P}}^2$ to lie on a conic. Namely, that the intersection points $\overline{ab}\cap \overline{de}$, $\overline{af}\cap \overline{dc}$, $\overline{ef}\cap \overline{bc}$ are aligned. One could ask an analogous question in higher dimension: is there a coordinate-free condition for $d+4$ points in ${\mathbb{P}}^d$ to lie on a degree $d$ rational normal curve? In this paper we find many of these conditions by writing in the Grassmann–Cayley algebra the defining equations of the parameter space of $d+4$-ordered points in ${\mathbb{P}}^d$ that lie on a rational normal curve. These equations were introduced and studied in a previous joint work of the authors with Giansiracusa and Moon. We conclude with an application in the case of seven points on a twisted cubic.

中文翻译：

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有理正态曲线的帕斯卡定理

帕斯卡定理给出了六个点的综合几何条件 $\mathrm{一种},\dots ,F$ 在 ${\mathbb{磷}}^2$躺在圆锥上。也就是说，交点$\overline{\mathrm{一种}乙}\cap \overline{d\mathrm{电子}}$, $\overline{\mathrm{一种}F}\cap \overline{dC}$, $\overline{\mathrm{电子}F}\cap \overline{乙C}$对齐。人们可以在更高维度上提出一个类似的问题：是否存在无坐标条件$d+4$ 点在 ${\mathbb{磷}}^d$ 躺在学位上 $d$理性正态曲线？在本文中，我们通过在 Grassmann-Cayley 代数中写入参数空间的定义方程，找到了许多这些条件$d+4$- 有序点 ${\mathbb{磷}}^d$位于理性正态曲线上。这些方程是在作者与 Giansiracusa 和 Moon 的先前联合工作中引入和研究的。我们以在扭曲立方体上有七个点的情况下的应用程序结束。