Timeline for Picard groups of (fiber) products
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 22, 2011 at 23:03 | comment | added | ACL | Exactly: for proper smooth connected $X$ and $Y$ over an algebraically closed field $Z$, $\mathop{\rm Pic}(X\times_Z Y)=\mathop{\rm Pic}(X)\oplus \mathop{\rm Pic}(Y)\oplus\mathop{\rm Hom}(A_X,P_Y)$, where $A_X$ is the Albanese variety of $X$ and $P_Y$ is the Picard variety of $Y$ (both being Abelian varieties of dimensions $h^1(X,\mathcal O_X)$ and $h^1(Y,\mathcal O_Y)$ respectively) | |
| Nov 22, 2011 at 19:36 | comment | added | Calc | Thank you very much, your examples are both very nice and simple. The first example makes me believe that the question, as I posed it, is really wrong. This because $H^2(X \times Y)$ contains $H^1(X) \otimes H^1(Y)$, and the latters could make algebraic classes on $X \times Y$. Does this make sense? | |
| Nov 22, 2011 at 19:31 | vote | accept | Calc | ||
| Nov 22, 2011 at 16:35 | history | edited | naf | CC BY-SA 3.0 |
Added answer for 3)
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| Nov 22, 2011 at 15:31 | history | answered | naf | CC BY-SA 3.0 |