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 $$m\mathbb {CP}^2\# n(-\mathbb {CP}^2)\ (m, n \geq 2)$$. The proofs of these results are given by studying a family of the Seiberg–Witten equations.

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