### Hurwitz monodromy, spin separation and higher levels of a Modular Tower, by Paul Bailey and Michael D. Fried

Abstract: Each finite $p$-perfect group $G$ ($p$ a prime) has a
universal central $p$-extension. For a perfect group these central
extensions come from its {\sl Schur multiplier\/}. Serre gave a
Stiefel-Whitney class approach to analyzing spin covers of alternating
groups ($p=2$) aimed at geometric covering space problems. This
included the regular version of the Inverse Galois Problem.
Every finite simple group with order divisible by $p$ has an infinite
string of perfect centerless group covers exhibiting nontrivial Schur
multipliers for the prime $p$. Sequences of moduli spaces of curves
attached to $G$ and $p$, called {\sl Modular Towers}, capture the
geometry of these many appearances of Schur multipliers in degeneration
phenomena of {\sl Harbater-Mumford cover representatives}. These modular
curve tower generalizations inspire conjectures akin to Serre's open
image theorem. This includes that at suitably high levels we expect no
rational points. Guided by two papers of Serre's, these cases reveal
common appearance of spin structures producing $\ theta$-nulls on these
moduli spaces. The results immediately apply to all the expected Inverse
Galois topics. This includes systematic exposure of moduli spaces having
points where the field of moduli is a field of definition and other
points where it is not.

Paul Bailey and Michael D. Fried <mfried@math.uci.edu>