The paper Stable maps and tautological classes of Faber-Pandharipande shows that the Gromov-Witten classes on spaces of relative stable maps of the projective line push forward to tautological classes on moduli spaces of stable curves.
It is claimed on p. 6 that $overline{M}_{g,0}(mu_1,ldots,mu_m)=overline{H}_g(mu_1,ldots,mu_m)$, that is, the space of relative stable maps with ramification profiles $mu_i$ over marked points of $mathbb{P}^1$ is equal to the space of admissible covers of a genus 0 curve. However, I am confused as to how this can be correct: if the degree is 1, then I believe the space on the right is empty, but the space on the left has many components in general, coming from starting with the identity map on $mathbb{P}^1$ and attaching ghost components of positive genus to the source. More generally, I don’t see how the presence of ghost components allow one to identify a stable map with an admissible cover.
Later Proposition 1 it is stated as a corollary the fundamental class of $overline{H}_g$ pushes forward to a tautological class on $overline{M}_{g,n}$, but because of the discrepancy above I don’t see why this is immediate.
Am I missing something?
