The Underlying Set of Product Variety in the Classic Language

Question

I'm really not familiar with product variety in classic algebraic geometry language(NOT SCHEME).

For $X, Y$ two prevarieties, we can define their product prevariety $X\times Y$ by gluing affine cases. My questions:

1. Is the underlying set of $X\times Y$ the same as the Cartesian product as sets? (I know the topology is not)

2. If $X, Y$ are varieties, is $X\times Y$ also variety?

3. Does the projection maps $p: X\times Y\rightarrow X$ have any property? For example, is it a closed map?

4. Let $i_{y_0}: X\rightarrow X\times Y$ be $x\mapsto (x,y_0)$ for a fixed $y_0\in Y$. Is this map a closed embedding?

I'm also wondering if there is any reference for algebraic geometry in classic language? Thank you in advance.

Answer 1

By "prevarieties" I assume you mean what others may call not-necessarily-separated algebraic varieties.

1. Yes, as a set it is the same as the product of the underlying sets.
2. Yes, it is shown in Kempf's "Algebraic Varieties", page 26. Basically it is shown by proving that a product of affine varieties is an affine variety, and then taking an open cover of products of affines from each variety's own affine cover.
3. It is not always closed, but it is open. This can be shown due to the fact that the projection satisfies the axioms of a quotient morphism, which is open.
4. It is closed and if I remember correctly it is in fact an isomorphism onto its image.

I share your frusturation in the search for extensive materials in classical algebraic geometry.