Some refined higher type adjunction inequalities on 4-manifolds

We further sharpen higher type adjunction inequalities of P. Ozsv'ath and Z. Szab'o on a 4-manifold $M$ with a nonzero Seiberg-Witten invariant for a Spin${}^c$ structure $\mathfrak{s}$, when an embedded surface $\mathrm{\Sigma }\subset M$ satisfies $[\mathrm{\Sigma }]\cdot [\mathrm{\Sigma }]\ge 0$ and $$\mathrm{\mid }⟨[\mathrm{\Sigma }],c_1(\mathfrak{s})⟩\mathrm{\mid }+[\mathrm{\Sigma }]\cdot [\mathrm{\Sigma }]\ge 2b_1(M)\mathrm{.}$$