Let $X$ be a smooth compact 4-manifold. Then every element of $H_2(X;\mathbb{Z})$ can be represented by a smooth embedded orientable surface and we have the so called genus function $G: H_2(X; \mathbb{Z}) \to \mathbb{Z}_{\geq 0}$ which assigns to a homology class the smallest genus of such a smooth surface needed to represent it. Suppose that $x$ is a nontorision element of $H_2(X; \mathbb{Z})$. Does the sequence $G(x), G(2x), G(3x),...$ limit to infinity? Can there be arbitrarily large zeroes? Is there always a limit?