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What is the definition of a geometrically connected curve?

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Seriously, the first hit on google gives you the answer... google.fr/search?q=geometrically+connected –  Lierre Nov 9 '12 at 13:10
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1 Answer 1

For a variety over a non-algebraically closed field, "geometrically connected" means connected over the algebraic closure.

As an example where this fails, note that the curve $x^2+1=0$ in $\mathbb{A}^2$ is connected over $\mathbb{Q}$, but not over $\mathbb{Q}[i]$, where is becomes $(x+i)(x-i)=0$, which is a union of two lines. Hence this curve is connected but not geometrically connected.

You can also use the same adjective for many other properties, so that you can talk about something being geoemtrically integral, geometrically rational, etc...

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I think your example doesn't work, at least in the projective plane, where any two lines meet in a point (in this case in the point $[0:0:1]\in\mathbb{P}^1(\mathbb{Q}[i])$). –  Qfwfq Nov 9 '12 at 13:01
Projective or not, the two lines in the example meet at $x=y=0$. For an example of a connected, non geometrically connected curve, better consider the affine curve over $\mathbb{Q}$ with affine ring $\mathbb{Q}(\sqrt{2})[x]$. –  Matthieu Romagny Nov 9 '12 at 13:32
Yes I realise now that I do not properly think through my example as I wrote it in a rush... I have edited it accordingly. –  Daniel Loughran Nov 9 '12 at 14:10
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