Let $f: X\times_S Y \to X$ be the projection morphism in the definition of fiber product.$ U\subset X\times_S Y$ be an open set. Does $f(U)$ contain a non-empty open set of $X$? I know this can be reduced to the affine case.
If it is not true in general, can we save it by adding extra condition such as :$X,Y$ are noetherian, integral, etc.?
The problem comes from the attempt to prove: when $X$ is a noetherian integral separated scheme which is regular in codimension one, then $X\times_{\operatorname{Spec}\mathbb{Z}}\operatorname{Spec}(\mathbb{Z}[t])$ is also integral.