ag.algebraic geometry - Differential of the Torelli morphism at the boundary

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.