Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many points.

Is it true then that $K_X = f^{*}K_Y$ ?

  • 3
    $\begingroup$ No, let $f$ be the blow-up of a point in smooth $Y$. Then $K_X=f^*K_{Y}+ E$ where $E$ is the exceptional divisor. See, for example, Debarre's book "Higher dimensional algebraic geometry" $\endgroup$ – SashaP Oct 25 '16 at 18:31
  • $\begingroup$ Sorry, I forgot to write that $f$ does not contract any divisor. $\endgroup$ – Jessica_90 Oct 25 '16 at 18:33
  • $\begingroup$ So, you mean biratonial map, aren't you? $\endgroup$ – SashaP Oct 25 '16 at 18:42
  • $\begingroup$ Exaclty, a birational map (not necessarily a morphism) not contracting any divisor. $\endgroup$ – Jessica_90 Oct 25 '16 at 18:47

There are several issues with this question.

One issue is that if $f$ is a rational map and not a morphism, then you have to say what you mean by $f^*$. Another issue is that if $K_Y$ is not at least $\mathbb Q$-Cartier, then $f^*K_Y$ doesn't make sense even if $f$ is a morphism. (More precisely, there is no good way to define $f^*K_Y$. In other words, $f^*$ can really only be defined for $\mathbb Q$-Cartier divisors).

I am also not entirely sure what "ordinary singularities" mean, but let me assume that it includes at least "normal". You kind of need that to talk about divisors anyway. I guess it might also imply that $K_Y$ is $\mathbb Q$-Cartier, in which case the second issue is resolved.

I guess you could do the following: If $X$ is normal, then $f$ is defined in codimension $1$, so let $U\subseteq X$ be a dense open set where $f$ is defined and $X\setminus U$ is at least codimension $2$ in $X$. If $f$ does not contract a divisor, then the exceptional set is also at least of codimension $2$, so we might as well assume that $f|_U$ is an isomorphism onto its image. So, you could define "$f^*K_Y$" as the "closure" of $K_U$ on $X$ which should indeed be $K_X$, but this feels more as a push-forward from $Y$ to $X$ than a pull-back, so I'd rather denote it (as it is done in the literature) by $f^{-1}_*K_Y$. But with that it is true that $f^{-1}_*K_Y\sim K_X$ and you don't even have to worry about $K_Y$ being $\mathbb Q$-Cartier.

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