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Let $W$ be a $2$-dimensional complex manifold and $C\subset W$ a compact complex curve (possibly singular). I would like to know a reference for the following fact: there exists a collection $\{V_j\}_{j=1}^n$ of simply connected open sets such that

  • $C\subset\bigcup V_j$
  • $V_j\cap C$ is connected for every $j$
  • if $V_j\cap V_k\neq \emptyset$, then $V_j\cap V_k\cap C\neq\emptyset$ and both $V_j\cap V_k$ and $V_j\cap V_k\cap C$ are connected.

I am quite ok with saying that such thing is well known for $C$ smooth, but what about singular curves? Does anyone know a suitable reference?

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  • $\begingroup$ Aren't the only problems at the nodes? Topologically, cusps don't exist, so whatever works for smooth surfaces works for topological surfaces as well (I would guess). $\endgroup$ – Marco Golla Apr 18 '15 at 21:43
  • $\begingroup$ Point is that I don't want to guess! Just searching for a reference. $\endgroup$ – Samuele Apr 18 '15 at 23:47

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