  For any open, connected and bounded set $\Omega \subseteq \mathbb C^m$, let
  $\mathcal A$ be a natural function algebra consisting of functions
holomorphic on $\Omega$. Let $\mathcal M$ be a Hilbert module over the algebra
$\mathcal A$ and $\mathcal M_0\subseteq \mathcal M$ be the submodule of
functions vanishing to order $k$ on a hypersurface $\mathcal Z \subseteq
\Omega$. Recently the authors have obtained an explicit complete set of unitary
invariants for the quotient module $\mathcal Q = \mathcal M \ominus \mathcal
M_0$ in the case of $k=2$. In this paper, we relate these invariants to
familiar notions from complex geometry. We also find a complete set of unitary
invariants for the general case. We discuss many concrete examples in this
setting. As an application of our equivalence results, we characterise certain
homogeneous Hilbert modules over the bi-disc algebra.
