(I’m particularly considering abelian surfaces and attribute 0).
How dangerous is the moduli stack of abelian varieties (with no polarization or degree construction)? Is it an Artin stack? DM (Deligne-Mumford) stack?
How dangerous is the is the stack of abelian varieties with full 2 degree construction (so with a foundation for $A$)?
Think about the maps from both 2) above or stack of principally polarized abelian varieties to 1) above. Are these maps easy, are the geometric fibers finite (ie, are there solely finitely many principal polarizations on an abelian selection)?
Neither moduli house is a stack as a result of each level has the automorphism $-1$ however the identical is true for the moduli stack of elliptic curves and that’s nonetheless a DM stack and never too dangerous.
Even in attribute 0, the CM locus is greater dimensional so the “particularly stacky” locus has excessive dimension however I do not understand how critical the issue is.
For the second query, whereas $-1$ fixes the 2 degree construction, I suppose a generic CM automorphism would not repair it so maybe the second stack could be very good, or atleast nearly as good as that of elliptic curves?