The Moduli Space of Rational Maps

Abstract

We construct the moduli space, Md, of degree d rational maps on ℙ1 in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for M3. Using this construction, we give a method for solving the following problems: (1) explicitly construct, from a moduli point P ∈ Md(k), a rational map ∅ with the given moduli; (2) find the field of definition of a rational map ∅. We work out the method in detail for the case d = 3