Hi,


I need a precise reference for the following fact, which is certainly well known, but I do not find any.


I consider the natural glueing map of pointed curves $\overline{M}\_{g\_1,n}\times \overline{M}\_{g\_2,n} \rightarrow \overline{M}\_{g\_1+g\_2+n-1}$. The pullback of the Hodge bundle such be equal to the direct sum of the two hodge bundles of the factors of the product plus n-1 trivial factors. where is this written properly?



pull back of hodge bundle via glueing map
I don't know a reference but here is an argument. It suffices to consider the case when we glue two points only.


Suppose that we are given a curve $C \to S$ (not necessarily connected) and two disjoint sections, $D$ is the result of gluing these together, and $\nu \colon C \to D$ the natural map. Then there is an exact sequence
$$ 0 \to O\_D \to \nu\_\ast O\_C \to O\_S \to 0$$
of sheaves on $D$. Push forward along $p \colon D \to S$. If $C$ has two components and the sections are on distinct components, we see that the first row in the long exact sequence is exact and $R^1 p\_\ast O\_D \cong R^1 p\_\ast O\_C$. If $C$ is connected we instead get a short exact sequence $0 \to O\_S \to R^1p\_\ast O\_D \to R^1p\_\ast O\_C \to 0$. Dualizing gives the result for the Hodge bundle. 


