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Every variety here is complex analytic, or complex algebraic if it solves anything.

Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ and an irreducible curve $(C,0)\subset(H,0)$, does a surface $(S,0)$ such that $H\cap S=C$ (set-theoretically, as germs) always exist? Are there general conditions that give this (such as $C$ being complete intersection)?

I do not care much about the structure of the varieties, except for the dimension.

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    $\begingroup$ No that does not always exist. Consider the germ at the vertex of a cone over a smooth plane cubic, and let the curve be the line through the vertex corresponding to a nontorsion point on the plane cubic (use one of the flex points as the origin). $\endgroup$
    – Jason Starr
    Commented Jan 14, 2023 at 21:43
  • $\begingroup$ @JasonStarr could you elaborate a bit more on why there is no surface $S$ as asked? Thanks. $\endgroup$
    – MathBug
    Commented Jan 16, 2023 at 10:03

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