Let $k$ be a field and $D:=\operatorname{Spec}(k[t]/(t^2)$ the scheme of dual numbers over $k$.
Then what is the fibre product $D \times_k D$ with itself over $k$? In other words, what is $\operatorname{Spec}(k[t]/(t^2) \otimes_k k[t]/(t^2)$ And how do line bundles over this scheme look like?