Let $X^4$ be a simply-connected closed smooth 4-manifold. Then every element $x \in H_2(X; \mathbb{Z})$ can be represented by an embedded orientable surface and the minimal genus of such a surface is called the genus of $x$, denoted $g(x)$. An element $x \in H_2(X; \mathbb{Z})$ is called characteristic if the reduction of $x$ to $H_2(X; \mathbb{Z})$ is Poincar'{e} dual to $w_2$.

Are there some methods for computing/lower-bounding the minimum genus amongst all characteristic homology classes in $H_2(X; \mathbb{Z})$?

I saw in this paper of Hamilton that the self-intersection number of characteristic homology classes of fixed genus can only be so small and my intuition is that the genus increases as the self intersection tends toward $+ \infty$ (and that this is even provable in some cases), so maybe that is somehow useful.