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Lemma: A crepant birational map between quasi-projective normal varieties with only terminal singularities is in fact small modifitacions, i.e. isomorphism in codimension 1.

So, if $f:X\dashrightarrow Y$ is a crepant birational map of smooth varieties and $p\in X$ a point, then $f|_{X-\{p\}} : X-\{ p\} \dashrightarrow Y-\{ f(p)\}$ is a crepant birational map between quasi-projective smooth varieties so is a small modification by the lemma above and then $f$ is a small modification.

Is it right or im missing something?

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  • $\begingroup$ I don't see why you're removing a point. As stated, the Lemma applies even before you remove the point. $\endgroup$
    – Hugh Thomas
    Mar 14, 2013 at 18:56
  • $\begingroup$ I didnt notice that. So i think that the question is solved. $\endgroup$
    – Joaquín Moraga
    Mar 15, 2013 at 16:23

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