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Summary We will prove that every differentiably embedded sphere with self-intersection -1 in a simply connected algebraic surface withpg>0 is homologous to an algebraic class. If the surface has a minimal model with Picard number 1 or |Kmin| contains a smooth curve, and eitherpg orKmin2 is even, then every such sphere is homologous to a (-1)-curve, as conjectured by Friedman and Morgan.

On the (?1)-curve conjecture of friedman and morgan