Families of algebraic spaces

Let U be a scheme. Let us define a family of d dimensions proper algebraic spaces over U to be a morphism X —> U from an algebraic space X to U which is flat, proper, locally of finite presentation, such that all geometric fibres are equidimensional of dimension d. Let Fam_d denote that full subcategory of the stack Spaces whose objects X/U are families of d dimensions proper algebraic spaces. Then as discussed in the preceding post we conclude that Fam_d is a stack over (Sch).

In this post I want to point out that for this to work out it is absolutely necessary that we work inside the category of algebraic spaces, and not with schemes. Let me start discussing the low dimensions.

[d = 0] It is a fact that any family X —> U of 0-dimensional proper algebraic spaces over a scheme U is automatically represented by a scheme. This follows from Proposition Tag 03XX.

[d = 1] Let X —> U be a family of 1-dimensional proper algebraic spaces over a scheme U. Then etale locally on U the space X is projective over U (in particular a scheme). But, even if you assume the fibres of X —> U are geometrically integral it is not the case that Zariski locally on U the space X is a scheme. An explicit example is the example of non-effective descent in Bosch-Lutkebohmert-Raynaud, Neron Models, Section 6.7 (since after all in Fam_1 we do have effective descent).

[d = 2] Here there are even examples of X —> U where all fibres are smooth projective surfaces, and U is a smooth curve, but the total space is an algebraic space and not a scheme. The examples comes from degenerating a general degree 513* surface in P^3 to a surface with a single node and doing a small resolution of the node on the total space (after performing a 2:1 base change). Moreover, there is no finite type, faithfully flat base change after which X becomes a scheme.

So you see that in order to do moduli of geometrically very interesting objects it is really convenient to work with algebraic spaces! In fact, if you don’t then you will not see all of the families that you want to see…

*Footnote: Degree 514 works also, and degree 21 too, and…

1 thought on “Families of algebraic spaces

  1. There is an example of a family of rational nodal curves over a nonsingular surface over the complex numbers whose total space is not a scheme. Namely, it is Example 2.3 of “The stack of rational curves” by Damiano Fulghesu (Arxiv preprint 0901.1201).

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