Arithmetic and geometry of cyclic covers

When presented with a hypersurface in projective space, a common trick is to pass to a cyclic cover of the projective space branched along the given hypersurface. In this talk I will discuss various stack-theoretic issues which arise when trying to perform this construction in a family, and give applications to finiteness results for hypersurfaces over $Q$ with good reduction outside a given set of primes. This is joint work with Ariyan Javanpeykar and Siddharth Mathur.