## Geometrically connected curve

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 at 13:10

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.
 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 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 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 at 14:10