Let $\mathcal{M}\_2$ be the moduli space of genus two curves and $\mathcal{A}\_2$ the moduli space of principally polarized abelian surfaces. Then the Abel-Jacobi map gives an open embedding $\mathcal{M}\_2 \hookrightarrow \mathcal{A}\_2$. My question is, for which compactification $\overline{\mathcal{A}}\_2$ of $\mathcal{A}\_2$ does the open embedding $\mathcal{M}\_2 \hookrightarrow \mathcal{A}\_2$ extend over the Delign-Mumford compactification $\overline{\mathcal{M}}\_2$? 



Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?
This map is usually called the Torelli map, not the Abel-Jacobi map. In any case, Mumford observed that a certain toroidal compactification of $\mathscr{A}\_g$ admits an extension of the Torelli map; the original reference is [this paper of Namikawa](http://link.springer.com/article/10.1007%2FBF01433145), I think. That paper doesn't give a very good moduli description of the map; luckily Alexeev does in [this paper](http://alpha.math.uga.edu/~valery/preprints/cjtm.pdf). I imagine everything can be made extremely concrete in genus 2, but I don't know a good reference for this.


