Timeline for Galois invariant Picard group elements
Current License: CC BY-SA 3.0
15 events
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| Dec 13, 2017 at 12:45 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 13, 2017 at 10:14 | comment | added | Jason Starr | @DanielLoughran. Yes, that is what I am claiming. For the change of variables, $[u,v,z]=[x-y\sqrt{a},x+y\sqrt{a},z]$, the closed subscheme is $\text{Zero}(uv-tz^2)\subset \mathbb{P}^2\times \mathbb{G}_m$. There is a (non-Galois-invariant) section of the projection to $\mathbb{G}_m$ given by $\text{Zero}(v,z)$. The invertible sheaf of that section is the unique ample generator of the Picard group of the Picard group of $X\times_{\text{Spec}\ k}\text{Spec}\ k(\sqrt{a})$. Thus, the invertible sheaf is Galois invariant, even though the divisor is not Galois invariant. | |
| Dec 13, 2017 at 5:15 | comment | added | Daniel Loughran | I'm just trying to make all this very explicit. A special case of your construction is the following. Let $a$ be a non-square in $k$. Then take $$X: x^2 -ay^2 = tz^2 \quad \subset \mathbb{P}^2 \times \mathbb{G}_m.$$ This is conic bundle over $\mathbb{G}_m$ with a rational point. There is no section over $k$, but are you claiming that the the class in the Picard group of a section over $k(\sqrt{a})$ is Galois invariant? | |
| Dec 12, 2017 at 21:33 | vote | accept | Daniel Loughran | ||
| Dec 12, 2017 at 19:24 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 18:41 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 18:35 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 18:30 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 14:17 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 13:21 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 13:13 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 12:41 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| Dec 12, 2017 at 12:25 | history | edited | Jason Starr | CC BY-SA 3.0 |
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| S Dec 12, 2017 at 10:19 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Dec 12, 2017 at 10:19 | history | made wiki | Post Made Community Wiki by Jason Starr |