The Moduli Space of Cubic Rational Maps

We construct the moduli space, $M_d$, of degree $d$ rational maps on $\mathbb{P}^1$ in terms of invariants of binary forms. We apply this construction to give explicit invariants and equations for $M_3$. Using classical invariant theory, we give solutions to the following problems: (1) explicitly construct, from a moduli point $P\in M_d(k)$, a rational map $\phi$ with the given moduli; (2) find a model for $\phi$ over the field of definition (i.e. explicit descent). We work out the method in detail for the cases $d=2,3$.