I am moving to the study of moduli of curves and I am looking at these notes where $\mathcal{M}_{g,n}$ is described as an orbifold. In order to define the tautological ring the notion of homology and cohomology of an orbifold is sketched (here for an orbifold I mean a space which locally looks like a quotient of an open set of $\mathbb{C}^n$ by a finite group - see notes def 1.15). In particular it is said that the homology and the cohomology of an orbifold $X$ is just the homology and cohomology of the underline topological space $\hat X$ (everything is tought with rational coefficient). However, " for convention" the homology class of an irreducible suborbifold $Y \subseteq X$ is $\frac{1}{|stab Y|} \cdot [ \hat Y]$ (where $[ \hat Y ]$ is the class ot the underline topological subspace). And here come the problems:
Consider some elementary example such as $\mathbb{Z}/k\mathbb{Z}$ acting on $\mathbb{P}^1(\mathbb{C})$: here the class of every point different from zero and $\infty$ is the same as the class on $\mathbb{P}^1(\mathbb{C})$ but with this convention it happens that $[0] = \frac{1}{k} [\hat 0]$ and the same for $\infty$. Is there a "geometric" reasons for these denominators other than remembering the action of $\mathbb{Z}/k\mathbb{Z}$? What does it means in this situation that the homology of this space is the same as the homology of the underline topological space?
On the other hand what is the definition of the stabilizer of an irreducible suborbifold? For me it is clear that by definition one can define the stabilizer of a point because one have the invariance of point stabilizer in every chart of the orbifold. But what about if the suborbifold lies in more than one chart?
If someone could also give some references that treats an introduction to the tautological ring of a moduli space of n-pointed genus-g curves in a base-level way it would be very appreciated.
Thanks so much