# The Degree 3 Case

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1975)

In this chapter we give a modified construction of a Viehweg-Zuo tower, which yields a CMCY family of 2-manifolds suitable for the construction of a Borcea-Voisin tower.

Let R 1 the desingularization of the weighted projective space P(2, 1, 1), which is obtained from blowing up the singular point. We start with the family C of curves given by
$$R^1 \supset V (y^3 - x_1 (x_1 - x_0) (x_1 - a_1 x_0) (x_1 - a_2 x_0)(x_1 - a_3 x_0)x_0) \rightarrow (a_1, a_2, a_3) \in {\mathcal M}_3.$$
This family has a dense set of CM fibers. Since the degree of these covers of P1 does not coincide with the sum of their branch indices, it is not possible to work with usual projective spaces. Thus we work with weighted projective spaces P(2, . . . , 2, 1, 1) resp., their desingularizations to obtain Calabi-Yau hypersurfaces and a tower of cyclic coverings similar to the construction of E. Viehweg and K. Zuo . For this construction we have to recall some facts and to make some preparations in Section 8.1. In Section 8.2 we give our modified version of the construction of Viehweg and Zuo, which yields a CMCY family of Calabi-Yau 2-manifolds. Let R 2 be the desingularization of the weighted projective space P(2, 2, 1, 1), which is obtained from blowing up the singular locus. The CMCY family of Calabi-Yau 2-manifolds is given by
$$R^2 \supset {\tilde V} (y^2_3 + y^3_1 - x_1 (x_1 - x_0) (x_1 - a_1 x_0) (x_1 - a_2 x_0)(x_1 - a_3 x_0)x_0) \rightarrow (a_1, a_2, a_3) \in {\mathcal M}_3.$$
We indicate some involutions of this family, which make it suitable for the construction of a Borcea-Voisin tower, in Section 8.3.

## Keywords

Singular Point Jacobian Matrix Commutative Diagram Galois Group Maximal Rank
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