Let $X$ be a scheme of $dim >1$ over a field $k$. Let $p \in X$ be a closed point.
Will there always be a curve i.e 1-dim subvariety passing through $p$? [If the answer is no, is it because of some pathological example due to the field being finite etc?]
Can something more extreme happen like there being a scheme of $dim >1$ with no curves as subvarieties (maybe over finite fields)?