The canonical covering maps from Hurwitz varieties to configuration varieties are important in algebraic geometry. The scheme-theoretic fiber above a rational point is commonly connected, in which case it is the spectrum of a Hurwitz number field. We study many examples of such maps and their fibers, finding number fields whose existence contradicts standard mass heuristics.

This work was partially supported by the Simons Foundation through grant #209472 and, in its final stages, by the National Science Foundation, through grant DMS-1601350.