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2 Made the question a bit clearer (for myself at least)

This question come comes from Huybrechts's book Complex Geometry, an An Introduction. In proposition 1.1.35, the author claim claims that if f $f$ is an irreducible holomorphic germ in $\mathcal{O}_{\mathbf{C}^n,0}$ at the origin , of $\mathbf{C}^n$, then for any z closed $z$ sufficiently close to the origin the induced holomorphic germ induced by $f$ in the local ring of $\mathbf{C}^n$ at z is irreducible.

But the proof only show shows the claim hold holds on the complement of a thin subset.The question is:

Question. Is the claim is true or false. Anyone can ? Can anyone give an answer or a reference?

1

# An irreducible germ of holomorphic function at origin is still irreducible around the origin?

This question come from Huybrechts's Complex Geometry, an Introduction. In proposition 1.1.35, the author claim that if f is an irreducible holomorphic germ at origin, then for any z closed to origin the induced holomorphic germ at z is irreducible. But the proof only show the claim hold on the complement of a thin subset. The question is: the claim is true or false. Anyone can give an answer or a reference?