Let $ C$ a point of $ \partial \bar{M}_{g,n}$ , corresponding to a curve with exactly one node $ p$ . Then the normal space of $ \partial \bar{M}_{g,n}$ inside $ \bar{M}_{g,n}$ at $ C$ is given by $ T_p(C_1)\otimes T_p(C_2)$ .
Let $ C$ be a singular stable $ n$ -pointed curve of genus $ g$ , i.e. a point of $ \partial \bar{H}_{g,n}$ , with $ r\geq 1$ nodes. What is the normal space of $ \partial \bar{H}_{g,n}$ inside $ \bar{H}_{g,n}$ at $ C$ ?
The reason I am interested in this is the following. I would like to apply the “method of test curves” to $ \bar{H}_{g,n}$ , and the only examples I have come from the case of $ \bar{M}_{g,n}$ . In this case, given a family of curves $ X\to B$ entirely contained in a certain divisor $ \Delta$ of $ \partial\bar{M}_{g,n}$ , to compute the degree of the line bundle associated to $ \Delta$ I would consider the normalization $ \widetilde{X}\to B$ , then base change until the singular components of dimension $ 1$ are disjoint, and then take the self-intersection of these component and add it to the self-intersection of the isolated nodes. Can the same procedure be carried out on $ \bar{H}_{g,n}$ , or this method does not work anymore?
