  This paper deals with two families of algebraic varieties arising from
applications. First, the k-factor model in statistics, consisting of n-times-n
covariance matrices of n observed Gaussian variables that are pairwise
independent given k hidden Gaussian variables. Second, chirality varieties
inspired by applications in chemistry. A point in such a chirality variety
records chirality measurements of all k-subsets among an n-set of ligands. Both
classes of varieties are given by a parameterisation, while for applications
having polynomial equations would be desirable. For instance, such equations
could be used to test whether a given point lies in the variety. We prove that
in a precise sense, which is different for the two classes of varieties, these
equations are finitely characterisable when k is fixed and n grows.
