Suppose $k$ is an algebraically closed field, and $X$, $Y$ are two normal varieties over $k$. Is the product $X \times Y$ necessarily still normal?



Does the fiber product of two normal varieties remain normal?
The answer is yes.


In general one can define a normal morphism of schemes $f:X \rightarrow Y$ to be a flat morphism such that for every $y \in Y$ the fibre over $y$ is geometrically normal.


Then we have the following theorem on normality and base change (see EGA Ch 2 IV 6.14.1)


Let $g: Y' \rightarrow Y$ be a normal morphism of locally noetherian schemes. Then for every normal $Y$-scheme $X$ the fibre product $X \times\_Y Y'$ is normal.


Over an algebraically closed field flatness and geometric normality reduce to just being normal so the result follows.


