  We prove that the factorization of Appell's generalized hypergeometric series
satisfying the so-called quadric property into a product of two Gauss'
hypergeometric functions has a geometric origin: we first construct a
generalized Kummer variety as minimal nonsingular model for a product-quotient
surface with only rational double points from a pair of superelliptic curves of
genus $2r-1$ with $r \in \mathbb{N}$. We then show that this generalized Kummer
variety is equipped with two fibrations with fibers of genus $2r-1$. When
periods of a holomorphic two-form over carefully crafted transcendental
two-cycles on the generalized Kummer variety are evaluated using either of the
two fibrations, the answer must be independent of the fibration and the
aforementioned family of special function identities is obtained. This family
of identities can be seen as a multivariate generalization of Clausen's
Formula. Interestingly, this paper's finding bridges Ernst Kummer's two
independent lines of research, algebraic transformations for the Gauss'
hypergeometric function and nodal surfaces of degree four in $\mathbb{P}^3$.
