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

Abstract

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 $K_{\min }^2\not\equiv 7(\mathrm{mod}8)$ (here $K_{\min }$ is the canonical class of the minimal model).

arXiv comment13 pages, LaTeX 2.09