Need a counterexample to show that Cl$(X\times Y)$ is not always same as Cl$(X) \oplus $Cl$(Y)$

Asked Jun 07 '16 at 03:14

Active Jun 07 '16 at 05:06

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Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by $\operatorname{Cl}(X)$.

Suppose $X$ and $Y$ be two quasi projective varieties.What is the counterexample to show that $\operatorname{Cl}(X\times Y)$ is not always same as $\operatorname{Cl}(X) \oplus \operatorname{Cl}(Y)$ ?

I know that the above result hold if one of the $X$ or $Y$ is $\mathbb P^n$ but i dont see any "natural" counterexample in general case.Any ideas?

edited Jun 07 '16 at 05:06

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asked Jun 07 '16 at 03:14

Arpit Kansal

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Hartshorne IV 4.10 gives an example. – Elle Najt Jun 07 '16 at 03:19
@AreaMan To be clear, you're referring to an exercise in that section. – Hoot Jun 07 '16 at 03:39
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Yes, that is what I mean. Thanks. Specifically, the claim in the exercise is that, for $X$ an elliptic curve, $Pic(X \times X)$ is an extension of the endomorphism ring of $X$ (as an elliptic curve) by $Pic(X) \times Pic(X)$. @hoot – Elle Najt Jun 07 '16 at 03:43
Cool. I like the mentioned Ex. V.1.6 as well, although I guess it's a little indirect. – Hoot Jun 07 '16 at 03:46
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Here is a nice class of examples – Georges Elencwajg Jun 09 '16 at 09:41
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Possible duplicate of Picard group of product of spaces – user26857 Aug 26 '16 at 14:34

Need a counterexample to show that Cl$(X\times Y)$ is not always same as Cl$(X) \oplus $Cl$(Y)$

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