What is the definition of a geometrically connected curve?
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put on hold as offtopic by Ricardo Andrade, Stefan Kohl, Felipe Voloch, Chris Godsil, Yemon Choi Oct 22 at 2:22This question appears to be offtopic. The users who voted to close gave this specific reason:



For a variety over a nonalgebraically 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 $(x+i)(xi)=0$, 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... 

