Let $\pi:X \to S$ be a morphism of schemes (I can assume that $\pi$ is sufficiently nice, e.g. proper and flat, but certainly not smooth).

Does there exist a scheme $I_{X/S}$ which parametrises the irreducible components of the fibres of $\pi$?

In the case where $S = \mathrm{Spec}(k)$ for some field $k$ and $X$ is reduced, it is possible to construct such a scheme by hand. One takes $I_{X/k}$ to be the product of the spectrum of the algebraic closure of $k$ in the function field of each irreducible component of $X$. This has the property that for any field extension $k \subset K$, the points $I_{X/k}(K)$ are in bijection with those irreducible components of $X$ over $\bar{k}$ which are actually defined over $K$.

I am looking for such a construction which works in greater generality. I'm naively hoping that this might be buried in EGA somewhere...

As a closely related example, note that it is quite easy to construct a scheme of connected components, at least when $X/S$ is proper. Namely, one simply takes the Stein factorisation of $\pi$ to obtain a finite scheme $S'/S$, whose fibres parametrise the connected components of the fibres of $\pi$ in a natural way.

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    $\begingroup$ If there is, it's not going to be separated: consider a smooth conic degenerating to the union of two lines. $\endgroup$ – Martin Bright Sep 30 '15 at 7:20
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    $\begingroup$ Perhaps you mean "irreducible components of geometric fibers of $\pi$", and it is unclear what exactly is meant by "parameterizes" since even in the case of connected components the sense of "parameterizes" is somewhat weak because the formation of Stein factorization rarely commutes with non-flat base change. Anyway, there is no such construction buried in EGA. $\endgroup$ – grghxy Sep 30 '15 at 7:21
  • $\begingroup$ You might read Raynaud's article on specialization of the Picard functor. For a specializing family of curves (as in the previous two comments), the nonseparated scheme of irreducible components does play a role in the analysis of the nonseparated scheme that is the closure of the zero section in the nonseparated Picard scheme. $\endgroup$ – Jason Starr Sep 30 '15 at 12:53
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    $\begingroup$ You may also have a look at Matthieu Romagny, Manuscripta 136, 1–32 (2011) (in the context of algebraic stacks). $\endgroup$ – Laurent Moret-Bailly Sep 30 '15 at 17:51
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    $\begingroup$ Re @LaurentMoret-Bailly's reference: Matthieu Romagny, "Composantes connexes et irréductibles en familles" (MR). $\endgroup$ – LSpice Jul 5 '16 at 18:15

There is a definition of a functor of irreducible components in that paper. If $\pi$ is finitely presented with geometrically reduced fibres, then the functor is representable by an étale algebraic space. As mentioned by Martin Bright, it is not separated in general; if it is, then it's a scheme (like all quasi-finite separated algebraic spaces).

  • $\begingroup$ Great, thanks! This seems to be exactly what I was after. $\endgroup$ – Daniel Loughran Oct 1 '15 at 9:05

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