I know that this question has been asked here at least twice, but all versions of it don't seem to clarify the specific part I'm confused at. To make things worse, Ueno's book seems to bypass this making me think I'm stuck at something trivial. So here it goes
Given a $k[t]/{<t^2>}$ - valued point $Φ$ on $X$, I'm having trouble understanding why the $x$ corresponding to the image of $<tmod(t^2)>$ is $k$-rational. $k$ is a field
Again I'm terribly sorry if this is trivial or has been asked before and I missed it.
A $k-$rational point of $X$ is given by a map $\mathrm{Spec} k \rightarrow X$, and we say that the image of the closed point $x$ is $k-$rational. In this case we have a map $\mathrm{Spec} k \hookrightarrow \mathrm{Spec} k[t]/(t^2) \rightarrow X$, where the first map is the inclusion of the closed point of $\mathrm{Spec} k[t]/(t^2)$. So we see that the image is $k-$rational.