Minimal genus for 4-manifolds with $b^+=1$

We derive an adjunction inequality for any smooth, closed, connected, oriented 4-manifold $X$ with $b^+=1$. This inequality depends only on the cohomology algebra and generalizes the inequality of Strle in the case of $b_1=0$. We demonstrate that the inequality is especially powerful when $2\tilde \chi+3\sigma\geq 0$, where $\tilde \chi$ is the modified Euler number taking account of the cup product on $H^1$.