Wild Hurwitz spaces and level structuresArithmetic Geometry and Representation Theory Research Seminar

by Michael Temkin

Tuesday Apr 21, 2026, 3:05 PM → 5:05 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

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 the non-archimedean
perspective on the same example, and in the second part I will tell about 
a work in progress of Hippold, where a (logarithmic) modular version of compactified 
Hurwitz space of degree p is constructed when only (p-1)! is invertible. In particular, 
this conceptually explains phenomena observed by Abramovich-Oort. Then
I will describe another outcome of the same ideas. It was observed by Abramovich-Oort 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.