What is the definition of a geometrically connected curve?
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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... 

