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?

  • 2
    $\begingroup$ 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. $\endgroup$ – ulrich Aug 9 at 4:22
  • $\begingroup$ 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? $\endgroup$ – Aaron Landesman Aug 9 at 4:52
  • 2
    $\begingroup$ 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. $\endgroup$ – Angelo Aug 9 at 8:00
  • $\begingroup$ 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. $\endgroup$ – Aaron Landesman Aug 10 at 3:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.