  We consider a number of examples of multiplier algebras on Hilbert spaces
associated to discs embedded into a complex ball in order to examine the
isomorphism problem for multiplier algebras on complete Nevanlinna-Pick
reproducing kernel Hilbert spaces. In particular, we exhibit uncountably many
discs in the ball of $\ell^2$ which are multiplier biholomorphic but have
non-isomorphic multiplier algebras. We also show that there are closed discs in
the ball of $\ell^2$ which are varieties, and examine their multiplier
algebras. In finite balls, we provide a counterpoint to a result of Alpay,
Putinar and Vinnikov by providing a proper rational biholomorphism of the disc
onto a variety $V$ in $\mathbb B_2$ such that the multiplier algebra is not all
of $H^\infty(V)$. We also show that the transversality property, which is one
of their hypotheses, is a consequence of the smoothness that they require.
