  A viable and still unproved conjecture states that, if $X$ is a smooth
algebraic surface and $C$ is a smooth algebraic curve in $X$, then $C$ realizes
the smallest possible genus amongst all smoothly embedded $2$-manifolds in its
homology class. A proof is announced here for this conjecture, for a large
class of surfaces $X$, under the assumption that the normal bundle of $C$ has
positive degree.
