Let $X$ be a normal projective variety and $f$ be a finite self map of $X$, which takes smooth locus to smooth locus. Let $Y$ be a projective resolution of $X$.

Question; Does the map $f$ induce a self map $g$ of $Y$ which commutes with the map $f$ and the resolution maps ?

  • 2
    $\begingroup$ If you really mean for $X$ to be an arbitrary resolution of $Y$, then no: Let $f:Y\to Y$ be an automorphism, and $X\to Y$ be blow up where the centre is not invariant under $f$. $\endgroup$ Feb 6, 2023 at 11:26
  • $\begingroup$ The resolution of Hironaka respects smooth morphisms, e.g., etale morphisms. This also holds for some of the later resolution algorithms. $\endgroup$ Feb 6, 2023 at 12:07
  • $\begingroup$ @Donu Arapura I..I have some more condition on $X$ and on the map. We can assume that $X$ has only terminal singularity and has Piard rank 1 and the map takes singular locus to singular locus. Now the question is ; does there exist a projective resolution with the above property? $\endgroup$
    – LAPRAS
    Feb 6, 2023 at 12:52
  • $\begingroup$ @Jason Starr ... I did not understand your comments. Could you please explain a little more? $\endgroup$
    – LAPRAS
    Feb 6, 2023 at 12:56
  • $\begingroup$ @Donu Arapura...sorry canonical singularity $\endgroup$
    – LAPRAS
    Feb 6, 2023 at 13:00


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