Other Examples and Variations

In this chapter we consider the automorphism groups of our examples of CMCY families. We want to find some new examples of CMCY families of n-manifolds as quotients by cyclic subgroups of these automorphism groups. By using [20], Lemma 3.16, d), one can easily determine the character of the action of these cyclic groups on the global sections of the canonical sheaves of the fibers. In this chapter we state this character with respect to the pull-back action.

In Section 9.1 we see that the CMCY family W of 2-manifolds given by

$$R^2 \supset \tilde{V} (y^3_2 + y^3_1 = x_1 (x_1 - x_0)(x_1 - a_1 x_0)(x_1 - a_2 x_0)(x_1 - a_3 x_0)x_0) \to (a_1, a_2, a_3) \epsilon \mathcal{M}_3$$

has a degree 3 quotient, which is birationally equivalent to a CMCY family of 2-manifolds. This quotient is also suitable for the construction of a Borcea-Voisin tower. By using degree 3 automorphisms of W → M ₃ and the Fermat curve F₃ of degree 3, we construct the CMCY families Q → M ₃ and R → M ₃ of 3-manifolds in Section 9.2.

In Section 9.3 we consider a subgroup of the automorphism group of the CMCY family C ₂→M ₁ given by

$$\mathcal{P}^3 \supset V (y^4_2 + y^4_1 + x_1 (x_1 - x_0)(x_1 - x_0)(x_1 - \lambda x_0)x_0) \to \lambda \epsilon \mathcal{M}_1.$$

We find some degree 2 quotients of this family, which are birationally equivalent to CMCY families of 2-manifolds. In Section 9.4 we see that these families have involutions suitable for the construction of Borcea-Voisin towers. We consider a larger subgroup of the automorphism group of C ₂ in Section 9.5.