Wild level structures and Hurwitz spacesSeminar Algebraic Geometry (SAG)

by Michael Temkin

Thursday Jun 25, 2026, 10:30 AM → 11:30 AM Europe/Berlin

Hurwitz moduli spaces of smooth (and stable) curves are classical and well studied objects when the order of ramification is invertible. Up to now very little was known about the wild case. In particular, Abramovich--Oort described in 2000 a natural model over Z (with 2 non-inverted) of the classical space of double covers of the projective line ramified at four points. Already this case turned out to be quite mysterious and lacked a modular interpretation. In particular, this space turned out to be the blowing up of the Katz-Mazur modular curve X(2) at the supersingular point.

In my talk I will discuss this example in some detail and then will outline two modern continuations of this story:

1. A work of Hippold, where a moduli space of degree-p covers was constructed over Z[1/(p−1)!], including the example of [AO], and was proved to be log smooth over (Z[1/(p−1)!],log(p)) in quite a few cases.
2. My work in progress where a new modular model over Z of the classical level-N modular curve is constructed. It classifies stable N2-pointed genus-1 curves with an action of (Z/NZ)2 acting transitively on the marked points and satisfying some natural restrictions which ensure liftability to an elliptic curve with marked N-torsion. The new modular model is obtained from Katz-Mazur model as the minimal normal modification which separates all ordinary branches at the supersingular points.