The following is an example from Gathmann's notes on algebraic geometry:
I am having problems with showing, rigorously, that $\tilde X$ is given by the prescribed equations. First, I do not get why the RHS is even a closed subset. Yes, it is given by the zero set of a polynomial equation- but the topology on $\mathbb{A}^2\times\mathbb{P}^1$ is given by the product topology, and I do not see how to show that a zero set of a polynomial equation is closed in this product topology. Second, why can't we have a smaller closed subset containing $\Gamma$?
Thanks in advance.