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Let $f:X\rightarrow Y$ be a birational morphism of projective varieties. Assume that $Pic(X)$ is a free abelian group generated by $n$ divisors $D_1,...,D_n$.

Under which hypothesis on $X$ and $Y$ is it true that then $Pic(Y)$ is the free abelian group generated by the divisors $D_i$ not contracted by $f$ ?

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  • $\begingroup$ What are your hypotheses on the singularities of $X$ and $Y$? You should at least assume that $Y$ is normal. $\endgroup$ – abx Nov 21 '16 at 15:33
  • $\begingroup$ Sure we may assume that $X$ is smooth and $Y$ is normal. But I guess this is not enough, take for instance Y a quadric cone of dimension two. $\endgroup$ – Jessica_90 Nov 21 '16 at 15:36
  • $\begingroup$ You should assume that $Y$ is locally factorial if you want this to hold. There are criteria for local factoriality in SGA 2, and there are other criteria that come out of the Minimal Model Program. $\endgroup$ – Jason Starr Nov 21 '16 at 17:52
  • $\begingroup$ Thanks. If $Y$ is singular just in points and the singularities comes from contractions via $f$ of smooth subvarieties of $X$ of codimension greater or equal than two do we know that $Y$ is locally factorial ? $\endgroup$ – Jessica_90 Nov 21 '16 at 20:56
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    $\begingroup$ @Jessica_90. "If $Y$ is singular just in points and the singularities come from contractions via $f$ of smooth subvarieties of $X$ of codimension greater or equal than two do we know that $Y$ is locally factorial?" If there is a contracted subvariety $Z$ of codimension greater than or equal to two, then you know that $Y$ is not locally factorial. I believe this result is originally due to Abhyankar (but I cannot find the reference). I learned this in Debarre's wonderful textbook. In the following link, it is Prop. 8.12, p. 85, math.ens.fr/~debarre/M2.pdf $\endgroup$ – Jason Starr Nov 22 '16 at 10:57
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OK, apparently this ended up too long for a comment.... (and to see the connection, read the comments above)


It's actually very easy to prove that if $Y$ is locally $\mathbb Q$-factorial (even a little weaker than locally factorial!), then for any point $y\in Y$ in the image of the exceptional set of any projective birational morphism $f:X\to Y$ that exceptional set has to contain a divisor whose image contains $y$, which is a bit weaker statement than that the exceptional set is of pure codimension $1$ (which I bet is what Debarre says in the reference given by Jason and it is also proved in Shafarevich's book), but it already proves Jason's comment (modulo assuming projective) that small (projective) morphisms produce not locally factorial singularities.

The proof (which is kind of fitting your setup) is as follows: Let $H$ be an effective Cartier divisor on $X$ which is not numerically trivial on a curve contracted to $y\in Y$ and consider $f_*H$; if $Y$ is locally $\mathbb Q$-factorial, then (perhaps after restricting to an open subset) some multiple of $f_*H$ is a Cartier divisor. Replacing $H$ with the same multiple we may assume that $f_*H$ is Cartier. Then $f^*f_*H$ is (numerically) trivial on any curve contracted by $f$ and hence by the choice of $H$, they cannot be equal. Therefore $f^*f_*H-H\neq 0$ is an (effective) exceptional divisor whose image contains $y$.

This proof actually shows that if $f$ is a small projective morphism (small means that the exceptional set has codimension at least $2$), then some of your generators on $X$ will likely not end up in the Picard group of $Y$.

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