How Coarse the Coarse Moduli Spaces for Curves Are!

Abstract

For a geometric object P over a field k of char, p, the field of moduli kp for P is, roughly speaking, defined by the infimum of the fields of definition of P. One can refer the details of the notion to S. Koizumi. The chapter presents the assumption that C is a complete non-singular curve of genus g (≥ 2) over k and let x be the point, corresponding to C, on the coarse moduli space Mg of curves of genus g. In char, p = 0, it is well-known that the residue class field k(x) at the point x coincides with the field of moduli kc for C. However, in positive characteristic case, these are prone to be different. This chapter discusses hyperelliptic curves with an automorphism of order p. It also presents various examples to illustrate Kodaira–Spencer map for a hyperellliptic curve.