Suppose we have a reduced curve $(C,0)$ in $(\mathbb{C}^n,0)$ with an isolated singularity in the origin, and we take the normalisation of $\mathcal{O}_{C,0}$, which should be isomorphic to $\mathbb{C}\{t\}$. Is it true that $\dim_{\mathbb{C}}(\mathbb{C}\{t\}/\mathcal{O}_{C,0}) < \infty$? For plane curve singularities this is true, but I needed puiseux parametrisation to show this, which doesnt exist for curves with higher embedding dimension, and I cant really find this result anywhere... Is it even true or this only holds for plane curve singularities? Any help and refference is appreciated.
2026-03-27 18:26:40.1774636000
Is the codimension of a space curve germ in its normalisation finite?
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