Excuses for any potential mis-phrasing as I am not a mathematician. I hope the question is clear and self-contained. Here it goes:
Assume we have two objects, a 3D surface $X$ and a 3D curve $Y$ of genera $g_X$ and $g_Y$ respectively. $Y$ can be closed and have self-crossings. Let's take the cartesian product of these two objects $(X,Y)\rightarrow X\times Y$ and let $g_{XY}$ denote the genus of the product object. Given $(X,Y,g_X,g_Y)$, would it be possible to speak of any upper/lower bounds on $g_{XY}$ without explicitly computing the object/surface in product space?