  Fixed a point O on a non-singular surface S and a complete mO-primary ideal I
in its local ring, the curves on the surface X obtained by blowing-up I are
studied in terms of the base points of I. Criteria for the principality of
these curves are obtained. New formulas for their multiplicity, intersection
numbers and order of singularity at the singularities of X are given. The
semigroup of branches going through a sandwiched singularity is effectively
determined, too.
