The lecture today began, "Let $X$ be a normal, proper, flat scheme over $\mathbb{Z}$, with $\dim X = 2$ and $X(\mathbb{C})$ is smooth" This lecture is way too advanced.
Wikipedia's definition of normal scheme is "that the local ring at the point is an integrally closed domain. "
I was unable to find the definition of "proper scheme" and could only find proper morphism
I could find flat morphism but not flat scheme
A smooth scheme is well approximated by affine space near any point. I only know about smooth manifolds.
Therefore I am reading that part of all of his reasoning will break down for any realistic example.
The only example I could think of is a circle $x^2 + y^2 - 1$ since it is an example for everything.