ScalingStacks

Note that a composition of morphisms of curves is a morphism of curves. Let $f:Z\to Z^{\prime}$ be a morphism of curves. We have the following statements.

We say that $f$ is strict if $f(Z_{u})$ is closed in $Z^{\prime}_{u}$ and $f(Z_{o})\subset Z^{\prime}_{o}$. Note that this implies that $f(Z_{u})$ is also open in $Z^{\prime}_{u}$.

Let $Z$ be a curve.

If $X$ is a subcurve of $Z$, then $X$ is a curve with $X_{o}=X\cap Z_{o}$, $X_{exc}\subset Z_{exc}$ and $\iota$ is defined on $C_{X}(z)$ as the restriction of $\iota$ on $C_{Z}(z)$, for $z\in X_{exc}$. Note that $X_{u}$ is open in $Z_{u}$.

Equivalently, a subspace $X$ of $Z$ is a subcurve if it is a curve, $X_{o}=X\cap Z_{o}$ and the inclusion map $X\to Z$ is a morphism of curves.

We define an equivalence relation on connected components of $Z-Z_{exc}$: it is the relation generated by $T\sim T^{\prime}$ if there is $z\in Z_{exc}\cap\overline{T}\cap\overline{T^{\prime}}$, $U$ a small open neighbourhood of $z$ and $L\in\pi_{0}(U-\{z\})$ such that $L\subset T$ and $\iota(L)\subset T^{\prime}$.

Let ${\mathcal{E}}$ be the set of equivalence classes of connected components of $Z-Z_{exc}$. Given $E\in{\mathcal{E}}$, let $Z_{E}=\bigcup_{T\in E}\overline{T}$. The subspaces $Z_{E}$ of $Z$ are called the components of $Z$.

A curve has only finitely many components, each of which is a closed subcurve.

If $Z$ is non-singular, then its components are its connected components.

The local structure of a curve is described as follows. Let $z\in Z$. There is an open neighbourhood $U$ of $z$ that is a subcurve of $Z$ and an isomorphism of curves $U\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}X,\ z\mapsto 0$, where $X\subset{\mathbf{C}}$ is one of the following:

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  ${\mathbf{R}}$ viewed as an unoriented manifold, if $z\in Z_{u}-\partial Z_{u}$

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  ${\mathbf{R}}$ where ${\mathbf{R}}_{\geq 0}$ is unoriented and ${\mathbf{R}}_{<0}$ has either of its two orientations, if $z\in\partial Z_{u}$

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  ${\mathbf{R}}$ viewed as an oriented manifold, if $z\in Z_{o}-Z_{exc}$

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  $\mathrm{St}(n_{z})$ if $z\in Z_{exc}$.