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If S is a noetherian scheme and π : Z → X a morphism of S-schemes, where X is proper over S and Z is quasi-projective over S, then the set-valued contravariant functor $\Pi_{Z/X/S}$ on locally noetherian S-schemes, which associates to any T the set of all sections of $π_T : Z_T → X_T$, is representable by an open subscheme of $Hilb_{Z/S}$. This is an exercise in Nitsure, "Construction of Hilbert and Quot Schemes", and the proof is similar to the construction of the scheme of morphisms "Mor". My questions are:

  • What is known about this scheme? It should be locally noetherian and quasiprojective, right? Can we say anything more?
  • Can this be generalized somewhat? For example, can we dispense with the "X proper" assumption?
  • Is there an analogous statement in the analytic setting? By wich I mean, assuming π : Z → X a morphism of complex varieties, where X is compact and Z quasi-projective, for instance.

References would be nice, even to FGA if that is the right place to look.

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Check out Martin Olsson's paper "Hom stacks and restriction of scalars" for what happens when you relax some hypotheses (but not the properness). In general, mapping spaces tend to become infinite dimensional when you don't have properness on the source. –  S. Carnahan Sep 23 '11 at 13:41
    
Thanks! I can see why it makes sense that they should become infinite dimensional; but they still tend to exist? Or can there be some problem turning them into -I don't know- stacky objects? Or other topological pathologies? –  quim Sep 24 '11 at 9:26
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