# 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, 2019 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, 2019 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, 2019 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, 2019 at 3:22

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.