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?