theory for contraction maps on curves

Question

Do you know any references for studying the maps $\bar{M}_{i,n}\rightarrow \bar{M}_{j,n-2}$ (moduli space of stable curves with marked points) where the map is given by identifying two marked points?

Thank you.

Answer 1

I think the map you mention is $\overline{M}_{g-1,n+2} \rightarrow \overline{M}_{g,n}$. We start from a stable curve of arithmetic genus $g-1$ with $n+2$ markings. We identify the markings $n+1,n+2$. As a result we get a stable curve of genus $g$ with $n$ markings. One technical reference with proofs is a paper by Knudsen: The projectivity of the moduli space of stable curves, II. There is more elementary treatment with examples in the article: the moduli space of curves and their tautological ring by Vakil.