  Let $(G, X)$ be a Shimura datum, let $\Omega$ be a connected component of
$X$, let $\Gamma$ be a congruence subgroup of $G(\mathbb{Q})^{+}$, and consider
the quotient map $q: \Omega \to S:=\Gamma \backslash \Omega$. Consider the
Harish-Chandra embedding $\Omega\subset\mathbb{C}^{N}$, where $N=\dim X$. We
prove two results that give geometric conditions which if satisfied by an
algebraic variety $V \subset \mathbb{C}^{N} \times S$, ensure that there is a
Zariski dense subset of $V$ of points of the form $(x,q(x))$.
