I have some questions about algebraic geometry which might be elementary and boring (sorry).
Let $R$ be a ring - say, an integral domain which is Noetherian. Let $X$ and $Y$ be $R$-varieties - that is, integral separated schemes of finite type over $R$. Let $f: X \to Y$ be a dominant morphism of $R$-varieties. Let $\beta: Y' \to Y$ be a proper, birational morphism.
- I believe that the fibre product $X \times_Y Y'$ can be reducible but does anyone know a simple example of this phenomenon?
- Is it true that the fibre product $X \times_Y Y'$ has a unique irreducible component $X'$ which dominates $Y'$?
- If the answer to the previous question is yes, is it true that the composition $X' \to X \times_Y Y' \to X$ is birational?