The invariant ring of degree-four rational maps on the projective line

Abstract

Let $k$ be an algebraically closed field of characteristic zero. Degree-four rational maps on $\mathbb{P}^1$, up to conjugation, correspond to pairs of binary forms $(F,G)\in V_5\oplus V_3$. The associated invariant ring is the joint invariant ring $\mathcal{R}_{5,3}=k[V_5\oplus V_3]^{\mathrm{SL}_2}$. We determine a minimal generating set for $\mathcal{R}_{5,3}$, consisting of fifty explicit joint transvectants of degrees at most $18$. As consequences we describe the null cone of $V_5\oplus V_3$, construct six explicit absolute invariants that generate the function field of the moduli space $\mathcal{M}_4^1$, and give an effective criterion for determining conjugacy of degree-four rational maps.