J-Holomorphic Curves in a Nef Class

Taubes established fundamental properties of $J$-holomorphic subvarieties in dimension $4$ in [8]. In this paper, we further investigate properties of reducible $J$-holomorphic subvarieties. We offer an upper bound of the total genus of a subvariety when the class of the subvariety is $J$-nef. For a spherical class, it has particularly strong consequences. It is shown that, for any tamed $J$, each irreducible component is a smooth rational curve. It might be even new when $J$ is integrable. We also completely classify configurations of maximal dimension. To prove these results, we treat subvarieties as weighted graphs and introduce several combinatorial moves.