Wild Hurwitz Moduli Spaces and Level Structures

Hurwitz moduli spaces of covers of curves of degree d are classical and well studied objects if one assumes that d! is invertible and hence no wild ramification phenomena occur. There were very few attempts to study the wild case. In the most important one Abramovich and Oort started with the classical space H_{2,1,0,4} of double covers of P^1 ramified at four points and (following an idea of Kontsevich and Pandariphande) described its schematic closure H in the space of stable maps over Z. The result over F_2 was both strange and informative, but lacked a modular interpretation.

In the first part of my talk I will describe the example of Abramovich-Oort and then tell about a work in progress of Hippold, where a (logarithmic) modular version of Hurwitz space of degree p is constructed when only (p-1)! is invertible. In particular, it conceptually explains phenomena observed by Abramovich-Oort. In the second part I will describe another outcome of the same ideas. It was observed by Abramovich-Ollson-Vistoli that H is the blowing up of the modular curve X(2). This is not a coincidence, and the same ideas can be used to refine the wild level structures of Drinfeld and construct modular interpretation of the minimal modifications of the curves X(p^n) which separate ordinary branches at any supersingular point. This is a very recent work in progress and the precise description of the obtained spaces is still to be found.