TITLE: Differential of the Torelli morphism at the boundary
QUESTION [9 upvotes]: Let consider the Torelli morphism $T:\mathcal{M}_g \rightarrow \mathcal{A}_g$, from the moduli space of curves of genus $g$ to the moduli space of principal polarized abelian varieties of dimension $g$, that maps a curve to its Jacobian. The differential of $T$ at a point $[C]$ is the natural map
$$H^1(C, T_C) \rightarrow Sym^2H^1(C, \mathcal{O}_C).$$
I know that $T$ can be extended to a map 
$$T:\bar{\mathcal{M}_g} \rightarrow \bar{\mathcal{A}_g}$$
from the Deligne-Mumford compactification of $\mathcal{M}_g$ to some compactification of $\mathcal{A}_g$.
I would like to know if there is a way to describe the differential of $T$ at a point representing a nodal curve. More specifically, how can we describe the deformations space of a semi-abelian variety and in particular of a generalized Jacobian variety?
Over $\mathbb{C}$, by computing the period matrix, one can show that the differential of $T$ has maximal rank at each point representing a nodal curve with non-hyperelliptic  normalization. I'm wondering if, perhaps, there is a more algebraic way to see it.

REPLY [2 votes]: As mathwonk said, Friedman and Smith has what you want although is still over C. For a more general result, stated algebraically and a connection with Serre-Tate deformation theory, see my paper with Coleman, also in Inv. Math. (1992).