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Given a geometrically integral affine variety $X:=\mathrm{Spec}(K[X_1,\ldots, X_n])/(f_1,\ldots, f_m)$ over a possibly imperfect field $K$, does there always exist an affine variety $\tilde{X}$ geometrically integral and smooth (not just regular) over $K$ with a dominating rational map $f:\tilde{X}\to X$? If not, would it help if $X$ has 'enough' rational points (say Zariski-dense in $X(K^\mathrm{alg})$), or if $\mathrm{dim}(X)$ is very small? Thank you!

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  • $\begingroup$ Just define $\widetilde{X}$ to be the basic open $D(f)$ in $X$ for $f$ any nonzero polynomial in the appropriate Fitting ideal of $\Omega_{X/K}$. $\endgroup$ Apr 17, 2015 at 14:01
  • $\begingroup$ @JasonStarr: Ah! You are right! I kept thinking that these were not necessarily affine. Of course I was being silly! Thanks! $\endgroup$ Apr 17, 2015 at 14:44

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