Calabi-Yau hypersurfaces in projective space with finite automorphism groups are of interest for their applications to mirror symmetry. In [Commun. Math. Phys. 280, No. 3, 675–725 (2008; Zbl 1158.14034)], C. Doran, B. Greene and S. Judes introduced one-parameter families of quintic threefolds with discrete symmetries in projective \(4\)-space and proved the equality of the Picard-Fuchs equations associated to their holomorphic \(3\)-forms.
The paper under review gives an easy proof of this equality by finding rational maps between such a family \(X_t\) and the family of mirror quintics \(M_t\). The main theorem states that these rational maps are in fact quotients by a finite group related to the automorphism group of the quintics. The equality of Picard-Fuchs equation then follows from the fact that the holomorphic \(3\)-form on \(M_t\) pulls back to a holomorphic \(3\)-form on \(X_t\).
The construction of the rational maps and their equivalence with quotient maps is also generalized to one-parameter families of Calabi-Yau hypersurfaces in projective \(n\)-space.