Under what conditions is a product of dominant morphisms dominant?

Question

Let $f:X\to S$ and $g:Y\to T$ be dominant morphisms. Then under what conditions does it hold that $f\times g$ is dominant? Is it enough to suppose that $X,Y$ are irreducible? I need it only in fairly restrictive conditions (e.g. smooth projective over an algebraically closed field), but still I wonder in what generality it holds, and can't seem to find any references for it.

Answer 1

you need some flatness condition. just look at the affine case: for integral affine schemes dominance is equivalent to the condition that $A\to B$ is injective. now if $A\to B$ is injective and $C\to D$ is injective there is no reason for $A\otimes C\to B\otimes D$ to be injective.

but if you are working over an algebraically closed field then all is fine: $f:X\to Y$ is dominant between integral schemes $X,Y$ if and only if there are some open affine subschemes of $X,Y$ like $U=Spec\, A,V=Spec\, B,f(U)\subset V$ such that the induced map between $A\to B$ is injective. by this and the fact that everything is flat over a field(and product of two integral scheme over an algebraically closed field remain Integral) you can deduce the dominance of product.

the problem is topological so you can deduce the same for irreducible schemes over a field form the integral case.