The K($π$,1)-property for marked curves over finite fields

Abstract

We investigate the K($π$,1)-property for p of smooth, marked curves (X,T) defined over finite fields of characteristic p. We prove that (X,T) has the K($π$,1)-property if X is affine and give positive and negative examples in the proper case. We also consider the unmarked proper case over a finite field of characteristic different to p.