  Every multiplier algebra of an irreducible complete Pick kernel arises as the
restriction algebra $\mv = \{f\big|_V : f \in \cM_d\}$, where $d$ is some
integer or $\infty$, $\cM_d$ is the multiplier algebra of the Drury-Arveson
space $H^2_d$, and $V$ is a subvariety of the unit ball. For finite $d$ it is
known that, under mild assumptions, every isomorphism between two such algebras
$\mv$ and $\mw$ is induced by a biholomorphism between $W$ and $V$. In this
paper we consider the converse, and obtain positive results in two directions.
The first deals with the case where $V$ is the proper image of a finite Riemann
surface. The second deals with the case where $V$ is a disjoint union of
varieties.
