  Given a smooth, irreducible, projective surface $S$, let $g(S)$ be the
minimum geometric genus of an irreducible curve that moves in a linear system
of positive dimension on $S$. We determine the value of this birational
invariant for a general surface of degree $d$ in $\mathbb P^3$ and give a bound
for $g(S)$ if $S$ is a general polarised K3 or abelian surface. As soon as this
note appeared on the math arxiv, David Stapleton kindly pointed out to the
author a paper by Ein and Lazarsfeld, in which in turn a paper by Konno was
cited. Unfortunately the author was not aware of these two papers which contain
results that strictly include the ones of this note. The author is very
grateful to David Stapleton.
