Let $k \subseteq K$ be a field extension and $X$ a $k-$algebraic variety. Is it true that the projection $X_K = X \times_{\mathrm{Spec}k} \mathrm{Spec} K \to X$ is surjective? Since I don't know what the underlying topological space of $X_K$ is, I am finding it difficult to do anything...
Another (possibly related) question: is it true that $\mathcal{O}_{X_K}(U_K) = \mathcal{O}_{X}(U) \otimes_k K$ for all open subsets $U \subseteq X$? I think this should hold for affine subsets, but I am not sure whether it is true in general...