Deformation theory and the tame fundamental group

Abstract:

Let U be a curve of genus g with $n + 1$ points deleted, defined over an algebraically closed field of characteristic $p \geqslant 0$. Then there exists a bijection between the Galois finite étale covers of U having degree prime to p, and the finite $p’$-groups on $n + 2g$ generators. This fact has been proven using analytic considerations; here we construct such a bijection algebraically. We do this by algebraizing an analytic construction of covers which uses Hurwitz families. The process of algebraization relies on a deformation theorem, which we prove using Artin’s Algebraization Theorem, and which allows the patching of local families into global families. That our construction provides the desired bijection is afterwards verified analytically.