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Let $(X,p)\subset \mathbb{C}^N$ be the germ of a normal surface singularity, is it true that for a general hyperplane section $H$ passing through $p$ the curve $X∩H$ does not have an embedded component supported at the point $p$?

If the answer is negative, is there a natural kind of singularities in which this condition holds?

Thanks in advance.

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If $X$ is a normal surface, then it is in particular Cohen-Macaulay, so for any hypersurface $H$, the intersection $X\cap H$ will also be Cohen-Macaulay and hence the unmixedness theorem holds, which implies that all the components of $X\cap H$ have the same dimension (in particular, it cannot have any embedded components).

So, you need less: for a surface, $S_2$ is enough.

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