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This change of point-ofview from subvarieties of P2 to maps to P2 may seem to be a somewhat irrelevant one since Td,g and Vd,g are bijective and both contain Ud,g as a dense open subvariety. However, these two spaces are not isomorphic as varieties. Essentially, Td,g normalizes Vd,g at points [C] corresponding to curves with cusps. All these observations generalize: if you’re interested you’ll find a longer discussion in [33]. 52) In the example above show that for (a, b) in ∆ but different from (0, 0) the curve Ca,b has a node at the double root (x, y) = (−3b 2a , 0) and that the composition of the normalization map ˜a,b ✲ Ca,b ⊂ P2 with the projection to the x-axis is simply π : C branched over (x, y) = (0, 3b a ).

The compactification Ag was historically the first such compactification to be constructed [136] and it remains known as the Satake compactification in honor of its discoverer. Taking the closure of Mg in Ag yields a compactification of Mg which we’ll denote by Mg and also refer to as the Satake compactification. Unfortunately, the Satake compactification isn’t modular. Recall that this means that Mg is not a moduli space for any moduli functor of curves that contains the moduli functor of smooth curves as an open subfunctor.

Either of the potential choices for the lifting above has order 4 on X: (x, y, λ) ✲(1 − x, ±iy, 1 − λ) ✲(x, −y, λ). In other words, while the square of such a candidate lifting would give an automorphism of Xλ , this automorphism would have to be nontrivial. A. Why do fine moduli spaces of curves not exist? 37 There are a number of approaches to dealing with the obstructions to the existence of fine moduli spaces due to automorphisms. To simplify, we’ll restrict our discussion to moduli problems of curves but all these techniques are more generally applicable.

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Abelian Varieties by D. Mumford


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