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I must admit that I know nothing about p-adic geometry, so the following question may be completely trivial.

Let $V\subset K^n$ be an affine algebraic variety. Let $D$ be a polydisk, and $F$ be an analytic map $f:D\to K^m$. Suppose that for every smooth point in $V\cap D$ the rank of $df|_V$ is not exceeding $k$. My goal is to estimate the dimension of fibers $V\cap f^{-1}(f(z))$. It's trivial that at smooth points the dimension of fibers is at least $n-k$. Is it also true for singular points?

If $K=\mathbb{C}$, this is follow from the semicontinuity of the fiber dimension (proved, for example, in Chirka "Complex analytic sets"). Is it also true for $K=\mathbb{C}_p$?

I am looking for the reference (preferably), proof, or some ideas.

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  • $\begingroup$ Probably your K is algebraically closed of characteristich 0 and $f|_{D\cap V}$ is assumed to have non-empty smooth locus that meets every irreducible component. Anyway, the theorem on semi-continuity of fiber dimension relative to the "analytic Zariski topology" for a map between rigid-analytic spaces does hold over any non-archimedean field, and is part of more far-reaching results on Zariski-topological properties (on source and/or target) for various loci for fibers of rigid-analytic morphisms in recent work of Antoine Ducros; see arxiv.org/pdf/1107.4259.pdf $\endgroup$ – nfdc23 Nov 21 '15 at 4:52

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