Subsection 7.2.1. (0PH4) · ScalingStacks

7.2.1. Definitions

We consider now partially oriented $1$ -dimensional spaces. We build the theory so that the unoriented part is a manifold, and morphisms are injective on the unoriented part.

0P96

Definition 7.2.1. We define a curve to be a $1$ -dimensional space $Z$ endowed with

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an open subset $Z_{o}$ containing $Z_{exc}$
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an orientation of $Z_{o}-Z_{exc}$ and
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a fixed-point free involution $\iota$ of $C_{Z}(z)$ for every $z\in Z_{exc}$

satisfying the following conditions:

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$\partial Z=\emptyset$
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$Z-Z_{o}$ has finitely many connected components, none of which are points
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given $z\in Z_{exc}$ , given $U$ a small open neighbourhood of $z$ in $Z_{o}$ , and given $L\in\pi_{0}(U-\{z\})$ , then $L\cup\iota(L)\cup\{z\}$ has an orientation extending the given orientations on $L$ and $\iota(L)$ .

We put $Z_{u}=Z-Z_{o}$ . Note that $\partial Z_{u}=Z_{u}\cap\overline{Z_{o}}$ . Given $z\in Z-Z_{exc}$ , we have $|C(z)|=2$ and we define $\iota$ as the unique non-trivial automorphism of $C(z)$ .

We denote by $Z^{\operatorname{opp}\nolimits}$ the opposite curve to $Z$ all of whose data coincides with that of $Z$ , except for $Z_{o}-Z_{exc}$ , whose orientation is reversed.

Fix $n\geq 1$ . The $1$ -dimensional space $Z=\operatorname{St}\nolimits(2n)$ (cf §7.1.1) can be endowed with a structure of curve by giving ${\mathbf{R}}e^{i\pi r/n}$ the orientation of ${\mathbf{R}}$ for $0\leq r<n$ and setting $Z_{o}=Z$ . The involution $\iota$ is defined by $\iota({\mathbf{R}}_{>0}e^{i\pi r/n})={\mathbf{R}}_{<0}e^{i\pi r/n}$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2