A compactification of the moduli space of self-maps of \mathbb{CP}^1 via stable maps

We present a new compactification M(d,n) of the moduli space of self-maps of \mathbb{CP}^1 of degree d with n markings. It is constructed via GIT from the stable maps moduli space \overline M_{0,n}(\mathbb{CP}^1 \times \mathbb{CP}^1, (1,d)). We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group.