Why do we need to treat proper schemes, not only projective schemes?

Question

I know that many complete varieties are projective: For example, complete curves, smooth complete surfaces, arithmetic surfaces, and abelian varieties are projective. And even for general proper schemes, Chow's lemma says these are "almost" projective. And I think that concrete objects what we want to study are projective (e.g., when we try to find solutions of an quation, we treat these as a projective vaiety).

As I said many varieties are projective in practice, however, sometimes algebraic geometers take time to generalize propositions to proper schemes.

So my question is, why do we need to treat proper schemes? Or equivalently, please give me examples of concrete proper non-projective schemes which is itself application.

Thank you very much!

Answer 1

It's not an unreasonable attitude to take that one doesn't care very much about complete non-projective varieties. For example, complete curves and surfaces are automatically projective.

However, it is often easier to prove that a scheme is complete than that it is projective -- for instance, if we only have a description of the scheme via its functor-of-points.

Another advantage is that the definition of a projective scheme (or projective morphism) involves extraneous choices (it involves embedding in $\mathbb{P}^n$ for some $n$) whereas the definition of a complete scheme (or more generally, proper morphism) does not.

This question on mathoverflow also gives an example of a more natural flavor than Hironaka's classical example, where they arise in higher-dimensional geometry: https://mathoverflow.net/questions/111504/do-complete-non-projective-varieties-arise-in-nature

Answer 2

According to this preprint on families of K3-surfaces, when continuing a family of smooth projective varieties on an open base the continued family may have singular fibers and may no longer be a projective family. As stated in the paper, under certain conditions projectivity is guaranteed after some operations called flops.

I am not sure if there are examples known where projectivity necessarily. But it seems that degenerating families are a good spot to look, if one wants to come across complete non-projective schemes in a natural setting.