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Suppose that $V\subset {\mathbb C}^n$ is an affine subvariety of codimension $p$. How does one prove that $V$ is regular (i.e., is a smooth manifold) at its generic points?

In view of the Jacobian test for regularity (which is just the implicit function theorem in this case), it suffices to show that there exist a point $x\in V$ and polynomials $f_1,...,f_p$ in the defining ideal $I$ of $V$ so that the derivatives $df_1,..., df_p$ are linearly independent at $x$. However, I do not see why such point and polynomials would exist.

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There are proofs in the standard texts. The proof in Harris' "First Course" seems to be close to what you want: the idea is first to reduce to the case of hypersurfaces (using the fact every irred. variety is birational to a hypersurface), then prove that case using the fact that if $f$ is irreducible, $df$ does not divide $f$ (unless deg $f=1$, but that case is trivial anyway). –  Asal Beag Dubh Jul 3 '13 at 20:33
    
Using scheme language this is trivial. –  Martin Brandenburg Jul 4 '13 at 8:12
    
@AsalBeagDubh: Thank you for the reference, I will take a look. –  studiosus Jul 4 '13 at 13:39
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@MartinBrandenburg: How does the scheme language help here? –  studiosus Jul 4 '13 at 13:39

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