# 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.

• The answer to this question seems to depend on your choice of compactification of $A_g$. You might want to consider log structures on your nodal curves, together with the moduli problem given in section 4.6 of Olsson's book Compactifying Moduli Spaces of Abelian Varieties. – S. Carnahan Jun 14 '10 at 6:19
• I'm sorry for my late reply. I think you are right, though that book is not really clear to me (I think I will need some more time to understand it). In the complex case I'm thinking to the Voronoi compactification of the moduli space of abelian variety. I know this is not canonical but this is not a real problem for my purposes. I'm wondering what is the right analogous in the non-complex case. – V M Jul 23 '10 at 13:30