  The algebras obtained as fixed points of the action of the cyclic group $Z_N$
on the coordinate algebra of the quantum disc are studied. These can be
understood as coordinate algebras of quantum or non-commutative cones. The
following observations are made. First, contrary to the classical situation,
the actions of $Z_N$ are free and the resulting algebras are homologically
smooth. Second, the quantum cone algebras admit differential calculi that have
all the characteristics of calculi on smooth complex curves. Third, the
corresponding volume forms are exact, indicating that the constructed algebras
describe manifolds with boundaries.
