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

But the proof only shows the claim holds on the complement of a thin subset.

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

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Can any op link this question with this MSE post? – Frank Science Apr 8 '15 at 11:21

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