Dsiclaimer: this is a very basic question, but many books just skip this crucial point.
Let $X$ be a smooth projective variety defined over a number field $K$. A closed point $x\in X$ is said rational if $k(x)=K$. Let $X(K)$ be the set of rational points of $X$.
What does it mean to check the ``density'' of $X(K)$? Dense where?
My guess is that we consider the base extension $X_{\mathbb C}:=X\times_{K}\mathbb C$ and then we check the density of $X(K)$ in $X_{\mathbb C}$. Is it correct?
A scheme $X$ has an underlying topological space $|X|$. When one asks whether the $k$-points are dense, one usually refers to density as a subset of $|X|$.