Campedelli surfaces with fundamental group of order 8

TL;DR

This paper proves that certain algebraic surfaces called Campedelli surfaces have a universal cover that is a complete intersection of four quadrics in projective six-space, and determines properties of their fundamental group.

Contribution

It establishes that the universal cover of a Campedelli surface is a complete intersection of four quadrics in P^6 and clarifies the structure of its fundamental group.

Findings

Universal cover is a complete intersection of four quadrics in P^6

Fundamental group cannot be the dihedral group of order 8

Provides a correction to previous incomplete manuscript

Abstract

We prove that an etale cover Y of degree 8 of a Campedelli surface S is a complete intersection of four quadrics in P^6, obtaining as a consequence that Y is the universal cover of S, the covering group G=Gal(Y/S)is the topological fundamental group of S and that G cannot be the dihedral group of order 8. This paper patches up an incomplete manuscript of the third author.