Universal Family Obstructed By Automorphisms

Example: Let \(\mathcal F\) be the moduli functor of smooth curves of genus \(g\), i.e. for a scheme \(S\) \(\mathcal F(S)\) is the set of isomophism classes of families of smooth genus \(g\) curves over \(S\) and for a morphism \(S\to T\) of schemes \(\mathcal F(T)\to \mathcal F(S)\) is given by pulling back families.

Recall that if there exists a family \(\mathscr E\to S\) which is non-trivial and has mutually isomorphic fibers, then \(F\) cannot be representable. Given an object \(E\in F(\mathbb C)\) with a non-trivial isomorphism we can glue together fibers using this automorphism to construct such a non-trivial family \(\mathscr E\) over \(S\).

To do this in the curves case, start with genus \(g\) curves \(C\to \Spec \mathbb C\) with a non-trivial automorphism \(\alpha\). We can then construt a non-trivial family \(\mathscr E\to S\) all of whose fibers are isomorphic to \(C\) in the following way:

1. Start with the trivial family \(\pi:C\times \mathbb P^1 \to \mathbb P^1\)
2. Glue \(0, \infty\in \mathbb P^1\) to obtain a map \(\mathbb P^1\to S\), where \(S\) is the nodal cubic
3. Glue the fibers \(\pi^{-1}(0)\) and \(\pi^{-1}(\infty)\) using the nontrivial automorphism \(\alpha\) to obtain a family \(\mathscr E\to S\).
4. This family cannot be pulled back from \(C\to \Spec \mathbb C\). if it were, then the projection map \(p: C\times \mathbb P^1\to C\) (the pullback of \(C\to \Spec \mathbb C\) along the structure map \(\mathbb P^1 \to \Spec \mathbb C\)) would factor as the composition of a map \(\mathscr E\to C\) and the gluing map \(C\times \mathbb P^1 \to \mathscr E\). However, the restriction of \(p\) to the fibers \(C_0,C_\infty\) yields an isomorphism