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.

Compactifying Moduli Spaces of Abelian Varieties. $\endgroup$