Moduli · Question 2.2

Consider \begin{align*} (1) &,, \text{stab}*([\overline{\mathcal{M}}^{,\sim}{g,n} (\mathbb{P}^1,a_i p_i)]^{\text{vir}});\ (2) &, ,\text{stab}*(\text{Adm}{g,n}(\mathbb{P}^1,a_ip_i)), \text{where Adm is the space of admissible covers;}\ (3) &, ,\sigma^(0), \text{where},, \sigma: \mathcal{M}_{g,n}^{ct}\rightarrow Pic^0_g ,,\text{is given by},,(C,p_1,..,p_n)\mapsto (C, \mathcal{O}(\Sigma a_ip_i)). \end{align} It is known that (1)=(3) on $\mathcal{M}_{g,n}^{\rm ct}$.

a) Find an expression for (1) and (2) in the tautological ring of $\overline{\mathcal{M}}_{g,n}$.

b) Does (3) extend to $\overline{\mathcal{M}}_{g,n}$?