# Implicit function theorem for singularities

I am looking for an implicit function theorem which holds also on singular spaces, at least if the singularities are "mild".

For example, let $0 = z^2 - x y + z w + w^2 + \epsilon w$ define a hypersurface in $\mathbb{C}^4$ with coordinates $(x,y,z,w)$ and a parameter $\epsilon \in \mathbb{C}$. Let $h \colon \mathbb{C}^4 \to \mathbb{C}$ be the projection to the last coordinate $w$. The restriction of $h$ to this hypersurface is not submersive in $0$ if $\epsilon \neq 0$, hence the implicit function theorem cannot be applied in $0$ for finding a (local, holomorphic) section of $h$.

However, for $\epsilon = 0$ the hypersurface is singular in $0$, and a (holomorphic) section $f \colon \mathbb{C} \to \mathbb{C}^4$ of $h$ is given by $\zeta \mapsto (3\zeta,\zeta,\zeta,\zeta)$. At least when looking at the Zariski tangent spaces, the map $h$ is actually submersive in $0$, and my question is whether the existence of the section $f$ of $h$ can be guaranteed locally by some version of the implicit function theorem, ideally also for other types of singularities.

• According to A'Campo, singularity theory is the branch of mathematics covering those rare cases where implicit function theorem does not work :) – Alex Degtyarev Feb 9 '15 at 21:39
• If seriously, similar things (maybe not quite what you want) are related to Milnor fibers, so you may want to check Milnor's book on singularities of complex hypersurfaces. – Alex Degtyarev Feb 9 '15 at 21:42