Starting from a curve $C$ (over $\mathbb C$, projective if you like, irreducible, etc), is there a somewhat canonical way of constructing a curve $\tilde C$ and a map $\tilde C\to C$ which only separates the branches of $C$ at singular points but which does not resolve the singularities in the branches themselves?
Separating branches of a curve but not fixing anything else
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algebraic-geometry