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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 $\overline{\mathcal{M}}_2$?

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  • $\begingroup$ I think Klaus Hulek and Sam Grushevsky had some results about this. $\endgroup$ – Sasha May 18 '15 at 17:09
  • $\begingroup$ Just a comment: Let $\tilde{\mathcal{M}_2} \subset \overline{\mathcal{M}_2}$ be the partial compactification obtained by adding stable curves whose Jacobian is compact. Then the Torelli map extends to an isomorphism $\tilde{T} : \tilde{\mathcal{M}_2} \rightarrow \mathcal{A}_2$. A nice discussion of various compactifications, especially Bainbridge compactification, can be found in section 11 of "Modular embedding of Teichmueller curves" by Martin Moeller and Don Zagier. $\endgroup$ – shehryar sikander Apr 23 '18 at 16:24
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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, I think. That paper doesn't give a very good moduli description of the map; luckily Alexeev does in this paper. I imagine everything can be made extremely concrete in genus 2, but I don't know a good reference for this.

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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.

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