  In [1] some quotients of one-parameter families of Calabi-Yau varieties are
related to the family of Mirror Quintics by using a construction due to Shioda.
In this paper, we generalize this construction to a wider class of varieties.
More specifically, let $A$ be an invertible matrix with non-negative integer
entries. We introduce varieties $X_A$ and $\overline{M}_A$ in weighted
projective space and in ${\mathbb P}^n$, respectively. The variety
$\overline{M}_A$ turns out to be a quotient of a Fermat variety by a finite
group. As a by-product, $X_A$ is a quotient of a Fermat variety and
$\overline{M}_A$ is a quotient of $X_A$ by a finite group. We apply this
construction to some families of Calabi-Yau manifolds in order to show their
birationality.
