Some refined higher type adjunction inequalities on 4-manifolds

We further sharpen higher type adjunction inequalities of P. Ozsváth and Z. Szabó on a 4-manifold $M$ with a nonzero Seiberg-Witten invariant for a Spin$^c$ structure $\frak{s}$, when an embedded surface $Σ\subset M$ satisfies $[Σ]\cdot [Σ]\geq 0$ and $|\langle [Σ],c_1(\frak{s})\rangle|+[Σ]\cdot [Σ]\geq 2b_1(M).$

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