Suppose $X$, $Y$, and $Z$ are smooth irreducible schemes, and $X \to Z$ and $Y \to Z$ are dominant maps.

I have a certain point $(x, y)$ in the fiber product $X \times_Z Y$; and I'd like to know that it lies in some component of the fiber product which dominates $Z$.

What condition on the differentials $df|_x : T_x X \to T_z Z$ and $df|_y : T_y Y \to T_z Z$ will allow me to verify this?

It's fairly easy to see that the surjectivity of $df|_x$ (or $dg|_y$) alone is enough to guarantee the desired conclusion --- I'm hoping here for something weaker that would still suffice. I'd by happy if the surjectivity of $df|_x + dg|_y$ suffices, for example; but if it does not, I would be interested in any other weaker condition too.

Suppose $X$, $Y$, and $Z$ are smooth irreducible schemes [EDIT: of finite type over an algebraically closed field of characteristic zero], and $X \to Z$ and $Y \to Z$ are dominant maps.

I have a certain point $(x, y)$ in the fiber product $X \times_Z Y$; and I'd like to know that it lies in some component of the fiber product which dominates $Z$.

What condition on the differentials $df|_x : T_x X \to T_z Z$ and $df|_y : T_y Y \to T_z Z$ will allow me to verify this?

It's fairly easy to see that the surjectivity of $df|_x$ (or $dg|_y$) alone is enough to guarantee the desired conclusion --- I'm hoping here for something weaker that would still suffice. I'd by happy if the surjectivity of $df|_x + dg|_y$ suffices, for example; but if it does not, I would be interested in any other weaker condition too.