We study the parameterized complexity of the dominating set problem in geometric intersection graphs. • In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that DOMINATING SET on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size).• In two and higher dimensions, we prove that DOMINATING SET is contained in W for intersection graphs of semi-algebraic sets with constant description complexity. So far this was only known for unit squares. Finally, we establish W-hardness for a large class of intersection graphs.