Lemma 33.6.7. Let $k$ be a field. Let $X$, $Y$ be schemes over $k$.
If $X$ is geometrically reduced at $x$, and $Y$ reduced, then $X \times _ k Y$ is reduced at every point lying over $x$.
If $X$ geometrically reduced over $k$ and $Y$ reduced. Then $X \times _ k Y$ is reduced.