TITLE: Extending the Abel-Jacobi map over the DM-compactification $\overline{\mathcal{M}}_2$?
QUESTION [7 upvotes]: 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 $\overline{\mathcal{M}}_2$?

REPLY [6 votes]: In genus two the situation is very simple. All toroidal compactifications of $A_2$ are isomorphic and the DM compactification $\overline M_2$ is a toroidal compactification. I don't know a good reference unfortunately.