Dihedral Galois covers of algebraic varieties and the simple cases

TL;DR

This paper studies dihedral covers of smooth algebraic varieties, providing a detailed description of their structure, invariants, and deformations, with applications to fundamental groups.

Contribution

It offers a comprehensive structure theorem for dihedral covers and introduces explicit classes of simple and almost simple dihedral covers with their invariants.

Findings

Described Weil divisors and Picard groups of double covers

Provided a structure theorem for dihedral covers

Determined invariants and deformations of simple dihedral covers

Abstract

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 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.