Timeline for Does "all points rational" imply "constant" for this "cubic" curve over an arbitrary field?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Jan 30, 2016 at 12:57 | vote | accept | Max Horn | ||
| Jun 8, 2011 at 20:04 | comment | added | Kevin Buzzard | Max -- I just raced down some comments. I'm not saying I've answered the question -- I hoped someone could either take over or point out a problem. If $K=F(\sqrt{d})$ then the moment I adjoin another $\sqrt{d}$ to the situation I can unravel everything. $LP=MN$ guarantees that I don't lose control at this point. That's the proof plan anyway. | |
| Jun 8, 2011 at 9:22 | comment | added | Max Horn | Accepting this as a black box for the moment, the rest seems clear, assuming new-$F$ maps diagonally into new-$K$. Indeed, can't we then immediately conclude $g=h$? | |
| Jun 8, 2011 at 9:21 | comment | added | Max Horn | In your answer, everything up to the final paragraph seems clear. But why exactly does $LP=MN$ imply you can change the setting as described? You somehow seem to arrive at the conclusion that the curve $(a(x,y),b(x,y))$ in the old $K$ (where multiplication "mixes" real and imaginary parts) can be ``untangled'' to $(g(x),h(y))$ and we can thus switch to $\overline F\oplus\overline F$. Since you mention it so casually, it's probably something terribly basic. But I don't see how it follows. Can you give me a hint? | |
| Jun 7, 2011 at 23:50 | comment | added | Kevin Buzzard | Over the algebraic closure it seems like I have something like a rational map from affine 2-space to affine 2-space, of the form $(s,t)\mapsto (g(s),h(t))$ (this is the statement that the map is defined over $K$), and whose image is contained in a diagonal affine 1-space. This means each horizontal line and each vertical line must be mapped to a point, and this means that the image of all of affine 2-space must be a point. Am I now done? | |
| Jun 7, 2011 at 23:16 | history | edited | Kevin Buzzard | CC BY-SA 3.0 |
typo
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| Jun 7, 2011 at 21:50 | history | answered | Kevin Buzzard | CC BY-SA 3.0 |