I am learning about fiber products at the moment and I was wondering about the following. Suppose I have affine varieties $X, Y$ over an algebraically closed field $k$. Say $X \subseteq \mathbb{A}^n$ and $Y \subseteq \mathbb{A}^m$, and I'm thinking as a subset of $k^n$ and $k^m$ respectively. Is it then the case that $X \times_k Y$ as a set equal to the Cartesian product of $X \times Y \subseteq \mathbb{A}^{n+m}$?
2026-03-25 19:06:53.1774465613
Is fiber product of two affine varieties the same as Cartesian product as sets?
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