  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 $\frak{s}$, when an embedded surface $\Sigma\subset M$
satisfies $[\Sigma]\cdot [\Sigma]\geq 0$ and $$|\langle
[\Sigma],c_1(\frak{s})\rangle|+[\Sigma]\cdot [\Sigma]\geq 2b_1(M).$$
