On the $(-1)$-curve conjecture of Friedman and Morgan

Main difference with previous version: we prove that every differentiably embedded sphere with self intersection $-1$ in a simply connected algebraic surface with $p_g >0$ is homologous to a $(-1)$-curve if $|K_{\min}|$ contains a smooth irreducible curve of genus at least 2 and $p_g$ is even or $K_{\min}^2 \not\equiv 7 \pmod8$ (here $K_{\min}$ is the canonical class of the minimal model).