仿射空间束在原点处的扎里斯基余切空间
The Zariski Cotangent Space at the Origin of the Pencil Diffeological Space
AI总结：
本文研究仿射空间束在原点处的扎里斯基余切空间，证明其由过原点直线的导数决定，还计算了相关切空间，发现奇异点局部光滑结构可产生新拓扑概念。
AI中文摘要：
仿射空间束是一种平面，其原点处的光滑结构仅由过原点的直线检测。我们证明原点处的扎里斯基余切空间完全由这些直线上的导数决定，且不同直线上的导数恰好构成一个σ-连续族（一种弱连续性形式）。我们还根据该描述计算了原点处的外切空间和右切空间，这表明仿射空间奇异点处的局部光滑结构可产生意想不到的拓扑概念。
英文摘要：
The pencil diffeological space is a plane whose smooth structure at the origin is detected only along lines through it. We show that the Zariski cotangent space at the origin is entirely determined by the derivatives along these lines. Moreover, the derivatives along different lines form precisely a $σ$-continuous family, a weak form of continuity. We also compute the external and right tangent spaces at the origin from this description. This suggests that the local smooth structure at a singular point of a diffeological space can give rise to unexpected topological notions.
