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In Stein Manifold and Holomorphic Mappings, by Forstnerič, I refer to Definition 8.9.9: An exposed point is a point belonging to a certain subset $\Sigma$ of $\Bbb C^2$, enjoying certain properties. This set $\Sigma$ is pointed as

Locally closed complex curve, possibly with boundary.

A complex curve is the holomorphic image of some open set $D\subseteq\Bbb C$ (including the whole $\Bbb C$).

Closed means that it coincides with its closure.

Calling $F\colon D\to\Bbb C^2$ holomorphic, we can extend it smoothly to the boundary $\overline D$ (in case $D\neq\Bbb C$) and define $\Sigma:=F(\overline D)$.

What does "locally" mean?

Why "possibly with boundary"? Does it have to have boundary or not?

Can somebody explain? Thanks

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    $\begingroup$ You can't always extend to the boundary smoothly. Locally means locally in $\mathbb{C}^2$, not locally in $D$. Possibly with boundary means parameterized locally by a closed half plane. $\endgroup$
    – Ben McKay
    Commented Sep 20, 2021 at 12:03

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