Bounds on genus and configurations of embedded surfaces in 4-manifolds

For several embedded surfaces with zero self-intersection number in 4-manifolds, we show that an adjunction-type genus bound holds for at least one of the surfaces under certain conditions. For example, we derive certain adjunction inequalities for surfaces embedded in $m$\mathbb{C}\mathbb{P}$^2# n(-$\mathbb{C}\mathbb{P}$^2)$($m, n \geq 2$). The proofs of these results are given by studying a family of Seiberg-Witten equations.