Consider the moduli space of stable curves \(M := \overline{M}_{g,n}(X, \beta)\) where \(X\) comes with the action of a torus \(T\) and \(F\subset M^T\) is a connected component of the torus fixed locus. To compute the weights of the \(T\) action on the normal bundle of \(F\), we apply the following strategy:
- compute the weights of the torus action on the BIG tangent space; i.e. the tangent space \(T_xM\) for \(x \in F\).
- find the moving part of \(T_xM\); the part not fixed by the torus action.
- the normal bundle \(N_{F/M}\), whatever it means in this case, should correspond to the moving part of \(T_xM\), since any tangent vectors which are part of \(T_xF\) “stay inside the fixed locus \(F\)” and thus should have weight 0.
This is basically the strategy followed. The hard part is describing the tangent space of a stack at a point explicitly enough to describe its weights. Alper has a description on the tangent space of a stack:
- See [Alp24, Definition 3.5.7] for a definition of tangent space.