Local discriminant variety

Question

I'm looking for good (as simple as it is possible) reference for the local discriminant variety.

I need it in the following situation: I have an unfolding $F: (\mathbb{K}^n \times \mathbb{K}^p, 0) \to (\mathbb{K}^n \times \mathbb{K}^p, 0) $ of a germ $f=f_0: (\mathbb{K}^n,0) \to (\mathbb{K}^n, 0) $ in the form $F(x, t)=(f_t(x), t)$. Here $\mathbb{K}= \mathbb{R}$ or $\mathbb{C}$, I don't need other fields, the germs $f$ and $F$ are analytic (or smooth). Let $\mathcal{D} \subset \mathbb{K}^p$ be the set of those $t $ parameter values for which $f_t$ has singular zeros, i.e. points $x \in \mathbb{K}^n $ with $f_t(x)=0$ and $ \det (\mbox{Jac}_x (f_t)) =0$. I'm looking for references for the fact that $\mathcal{D}$ is a local analytic (or smooth) variety.

I also ask for an advice for the terminology: Is $\mathcal{D}$ called local discriminant variety? For a fix $t$ can I call $f_t$ the perturbation of $f$ corresponding to the parameter value $t$, or does it have another name?

Additional information

• $\mathcal{D}$ is the generalization of the discriminant of 1-variable polynomials in the following way: $n=1$, $f=x^{p-1}$, and $t$ is the vector of the coefficients of a degree-$p-1$ polynomial $f_t$. $\mathcal{D}$ defined above is the zero set of the discriminant of $f_t$, i.e. the resultant of $f_t$ and $f'_t$.
• A multi-version for polynomials called discriminant variety can be found in the Maple help with references.
• On the other hand in a new book Singularities of mappings the notion of the discriminant is used in another context, namely for the set of the singular values of a map.

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Actually I need only for the fact that $\mathcal{D}$ has at least 1 complex codimension in the complex case. I guess I can prove it. The set $\mathcal{S}$ of singular points of $F$ in the source is the zero set of $\det (\mbox{Jac} (F))$, hence $\mathcal{S} $ has 1 complex codimension in $\mathbb{C}^n \times \mathbb{C}^p$. Its image $F(\mathcal{S})$ is the set of singular values of $F$, i.e. the discriminant in the sense of the Mond-Nuno-Ballesteros book. It has at least 1 complex codimension. Then $\mathcal{D}=F(\mathcal{S}) \cap \{x=0\} \subset \mathbb{C}^p$. I guess that the intersection is transverse, therefore the codimension is at least 1. Anyway, this argument does not answer my original question, just its original motivation.

Answer 1

In the complex case $\mathcal D$ is analytic when the zero locus of $f_0$ has isolated critical point. The set germ $S$ of pairs $(x,t)$ in $\mathbb C^n\times\mathbb C^p$ such that $f_t(x)=0$ and $\det(d_xf_t)=0$ is analytic. The discriminant $\mathcal D$ is the image of $S$ by the projection $\pi(x,t)=t$. Since $S\cap\{t=0\}=\{0\}$ (as set germs), the projection $\pi:S\to \mathbb C^p$ is finite and hence its image is analytic by the Remmert's Finite Mapping Theorem.