# Does the compactified Torelli map extend to a proper map of stacks?

Let $$M_g^{ct}$$ denote the moduli stack of compact type genus $$g$$ stable curves and $$A_g$$ the moduli stack of principally polarized $$g$$-dimensional abelian varieties.

Can someone provide a reference for the fact that the Torelli map extends to a map $$M_g^{ct} \to A_g$$ which is proper?

Here are some comments: This is stated, for example, in Section 1.3 of "The Torelli locus and special subvarieties" by Ben Moonen and Frans Oort. I believe I have found a reference, but am not completely certain if I'm interpreting things correctly. In "Compactified Jacobians and the Torelli Map" by Alexeev http://alpha.math.uga.edu/~valery/preprints/cjtm.pdf Corollary 5.4, it is stated that there are compactifications $$\overline{M}_g$$ and $$\overline{A}_g$$ such that the Torelli map extends to a map of their coarse spaces. I believe the constructed map actually defines a map on stacks (and not just coarse spaces) which would imply the map $$M_g^{ct} \to A_g$$ (which is the restriction of $$\overline{M}_g \to \overline{A}_g$$) is also proper. So it would be enough to answer the following question affirmatively: Is it correct that the map in Corollary 5.4 is actually coming from a map of stacks and not just a map of coarse spaces?

• It should be easy to prove properness using the valuative criterion (for stacks): A curve over a dvr has semi-stable reduction after a base change and then it is easy to see that the special fibre is of compact type iff the Jacobian has good reduction.
– naf
Aug 9 '19 at 4:22
• Thanks! Do you know why there is a map at all? Given the universal family of curves $C \to M_g^{ct}$, we can form $J := Pic^0_{C/M_g^{ct}}$, and we want to construct a polarization on J. So, we want a symmetric map $\phi: J \to \hat{J}$ with $(1, \phi)^* P_J$ ample, for $P_J$ the Poincare bundle on $J \times \hat{J}$. We have a polarization over $M_g \subset M_g^{ct}$ and Martin Olsson explained to me how to extend the isomorphism to $M_g^{ct}$ using finiteness of the components of Hom(J, J^t) via rigidity, Weil's extension theorem, and normality of $M_g^{ct}$. But how do you check ampleness? Aug 9 '19 at 4:52
• The Jacobian of a curve of compact type is the product of the Jacobians of the components, and the polarization is the product of the polarizations. This makes ampleness clear, I think. Aug 9 '19 at 8:00
• Ah, thanks so much; the above comments answer the question! What was confusing me about ampleness is that the extension from my previous comment was non-explicit. But I guess you can just explicitly check that effective degree g-1 line bundles in Pic^{g-1} parameterize a relative flat divisor, for example by checking it over DVRs, so it must be the one I was defining. I'm still curious to know whether Alexeev's result mentioned in the question applies to stacks, but I suppose that's more of a curiosity for the purpose of the original question. Aug 10 '19 at 3:22

## 1 Answer

I wrote an answer to the question above on my website. It is currently available at http://web.stanford.edu/~aaronlan/assets/properness-of-compact-type.pdf.

The main points were explained in the comments above, but let me say them again. One can first extend the usual Torelli map by the general fact that over a normal noetherian base, principal polarizations uniquely extend from a dense open to the whole base. The key point here is that ampleness is (perhaps surprisingly) a closed condition. This can be deduced from the Riemann-Roch theorem and vanishing theorem for abelian varieties (cf. p. 159 of Mumford's book on Abelian varieties).

Then, to verify properness, one can use the valuative criterion. The key inputs in applying the valuative criterion are properness of the Deligne Mumford compactification $$\overline{M_g}$$, and the fact that a smooth separated semi-abelian group scheme over a the spectrum of a dvr is necessarily an open subscheme of its Neron model. In a little more detail, given a family of abelian varieties over the spectrum of a dvr $$T$$, if the generic fiber is the Jacobian of a curve $$C$$, we want to show that after extending $$T$$ to some spectrum of a dvr $$T'$$, the whole family is the Jacobian of a relative compact type curve over $$T'$$. We can find such a stable curve by properness of $$\overline{M_g}$$. We have to check this is actually a compact type curve. However, its Jacobian is proper using the above fact, and so it must be a compact type curve. Finally, its principally polarized Jacobian agrees with the ppav over $$T'$$ because they have the same generic fiber.