  This is a quick survey on the characteristic varieties associated to rank one
local systems on a smooth, irreducible, quasi-projective complex variety $M$. A
key new result is Proposition 1.8, giving additional information on the
constructible sheaf $\F=R^0f_*(\LL)$, where $\LL$ is a rank one local system on
$M$ and $f:M \to S$ is a surjective morphism with $S$ a smooth curve and the
generic fiber $F$ of $f$ connected. Corollary 1.9 says that for $S$ compact,
the singular support of $\F$ cannot be a singleton.
