Configuration spaces for wildly ramified covers. (English) Zbl 1106.14301

Fried, Michael D. (ed.) et al., Arithmetic fundamental groups and noncommutative algebra. Proceedings of the 1999 von Neumann conference, Berkeley, CA, USA, August 16–27, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2036-2/hbk). Proc. Symp. Pure Math. 70, 353-376 (2002).
Let \(f: Y\to \mathbb{P}^1\) be a finite separable map over an algebraically closed field of characteristic \(p>0\). Let \(y\) be a ramification point of \(f\) and denote by \(x\) the image of \(y\) in \(\mathbb P^1\). In this paper, the authors associate a ramification invariant \(\mathcal R\) to the extension of complete local rings \(\widehat{\mathcal O}_{y}/\widehat{\mathcal O}_{x}\). If \(f\) is a (branched) Galois cover, the ramification invariant \(\mathcal R\) encodes the filtration of higher ramification groups of the decomposition group of \(y\). The ramification invariant appears to be a finer invariant, but it is not clear what extra information it encodes. The main difference between the ramification invariant \(\mathcal R\) and the filtration of higher ramification groups is that \(\mathcal R\) is also defined in the non-Galois case. It would be interesting to see what properties of the filtration of higher ramification groups extend to the ramification invariants. For example, whether there is an easy formula for the genus of \(Y\) in terms of the ramification invariants of the ramification points, as in the Galois case.
In the second part of the paper, the authors use the ramification invariant to define an analog to the configuration space in characteristic zero. The idea is as follows. Let \(\mathcal H\) be a Hurwitz space over the complex numbers, parameterizing covers of the projective line branched at \(r\) points with a fixed monodromy group and suitable prescribed ramification. Then \(\mathcal H\) is a finite cover of the configuration space \({\mathcal U}_r\), parameterizing \(r\) tuples of points in \(\mathbb P^1\). Hurwitz spaces have many applications to Galois theory [M. D. Fried and H. Völklein, Math. Ann. 290, No. 4, 771–800 (1991; Zbl 0763.12004)].
In positive characteristic, it is still possible to define Hurwitz spaces, but the map to the configuration space \({\mathcal U}_r\) is no longer finite. The ramification invariant allows one to define a space \({\mathcal P}({\mathcal R})\) which may replace \({\mathcal U}_r\) in positive characteristic. In the last section, the authors propose possible applications to Galois theory in positive characteristic.
For the entire collection see [Zbl 0993.00031].


14G32 Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory)
12F10 Separable extensions, Galois theory
14H30 Coverings of curves, fundamental group


Zbl 0763.12004