There is a strong interest in superelliptic and p-gonal surfaces. Historically, the surfaces, especially hyperelliptic surfaces, were linked to the study of certain integrals. Most recently, they are of interest in cryptography and fields of moduli. Among all surfaces, p-gonal surfaces are the surfaces with the simplest and most tractable equations. In this brief article we talk about normal forms of p-gonal surfaces, their automorphisms, and very briefly about families of p-gonal surfaces. Space does not allow discussion of applications to cryptography or fields of definition of superelliptic surfaces.
Contemporary Mathematics Volume 629, 2014 http://dx.doi.org/10.1090/conm/629/12570