Rational covering maps of the projective line (${\dim }_{\mathbb{C}}{\mathrm{Rat}}_m=2m+1$)

A degree-$m$ rational map of the complex projective line is represented by two degree-at-most-$m$ polynomials without a common root, modulo common scalar multiplication. This is an open set of ${\mathbb{CP}}^{2m+1}$. Quotienting by domain Möbius transformations subtracts three complex dimensions, giving real dimension $4m-4$ for degree $m\ge 2$. These are the local covering parameters of a multiple-covered J-holomorphic sphere.