Dihedral Galois covers of algebraic varieties and the simple cases

In this article we investigate the algebra and geometry of dihedral covers of smooth algebraic varieties. To this aim we first describe the Weil divisors and the Picard group of divisorial sheaves on normal double covers. Then we provide a structure theorem for dihedral covers, that is, given a smooth variety Y, we describe the algebraic “building data” on Y which are equivalent to the existence of such covers π:X→Y. We introduce then two special very explicit classes of dihedral covers: the simple and the almost simple dihedral covers, and we determine their basic invariants. For the simple dihedral covers we also determine their natural deformations. In the last section we give an application to fundamental groups.