When are valuative criteria useful?

Question

We have valuative criteria for properness and universal closedness of morphisms of schemes. I would agree that these criteria shed some light on the geometric nature of these morphisms. However, are there any situations in which these criteria are more convenient to use than simply starting from the basic definitions?

Unfortunately, all the applications I have seen so far felt like "Surely I could argue from the definitions themselves but the word "valuative" is so fancy and makes me sound smart so I am going to use it anyway."

I think that once you become experienced enough, the difference between different approaches might seem negligible; when giving examples assume you are a person who just finished studying Hartshorne (so your intuition is more-or-less at that level).

Answer 1

One very important point, implicit in the comments by @zzy and @abx, is that valuative criteria allow to check a property (e.g. properness) of (say) an $S$-scheme $X$ directly in terms of the functor of points of $X$. This is of course especially convenient if $X$ is defined by this functor, classical examples being Picard, Quot or various moduli functors. For these, properness (when applicable) can then be checked even before the functor is proved to be representable by a scheme. Moreover, in general the checking is straightforward, or even trivial.

For instance, let $V$ be a proper, smooth and geometrically connected variety over a field $k$ (algebraically closed, if you wish). To check that the Picard functor $\underline{\mathrm{Pic}}_{V/k}$ satisfies the valuative criterion for properness is a simple exercise on extending line bundles over a regular scheme. In contrast, I can't think of an easy way to prove directly that each component of $\underline{\mathrm{Pic}}_{V/k}$ is universally closed over $\mathrm{Spec}(k)$. (Note that in general $\underline{\mathrm{Pic}}_{V/k}$ is not of finite type, hence not proper, but its components are).

In the case of the moduli of curves (Deligne-Mumford) properness is not trivial but is obtained as a trivial consequence of a deep theorem, namely the semistable reduction theorem.

Answer 2

I'm not sure if you're looking only for examples where checking properness/separatedness/etc. is easier using valuative criteria; this is an example where proving something is made easier. The example is liftability of arcs under proper birational morphisms.

A bit of setup: Say $X$ is a variety over a field $k$; an arc $\gamma$ on $X$ is a morphism $\gamma: \mathrm{Spec}\, k[[t]] \to X$. Write $X_\infty$ for the set of arcs on $X$ (in fact, $X_\infty$ carries a natural scheme structure, but we ignore this). The set of arcs on $X$ carries a great deal of information about the singularities of $X$, so it's of interest in birational geometry to understand how sets of arcs behave under proper birational maps; in particular, given $f:Y\to X$ a proper birational map, we want to know what the relation between $Y_\infty$ and $X_\infty$ is. Say that $f$ is an isomorphism over $X-Z$ for some closed subset $Z$ of $X$. "Most" arcs on $X$ don't have scheme-theoretic image contained in $Z_\infty$ (i.e., the morphism $\mathrm{Spec}\, k[[t]] \to X$ doesn't factor through $Z$). Write $Z_\infty$ for the set of arcs on $X$ that do factor through $Z$, and likewise $(f^{-1}(Z))_\infty$ for the arcs on $Y$ factoring through the (set-theoretic) preimage $f^{-1}(Z)$.

We claim that the proper birational morphism $f:Y\to X$ induces a bijection between $Y_\infty-(f^{-1}(Z))_\infty$ and $X_\infty-Z_\infty$, which should be thought of as a bijection away from a "measure zero" set. Using the valuative criterion for properness though, this is super easy to see: take $\gamma $ in $X_\infty-Z_\infty$. Since $\gamma$ is not in $Z_\infty$ the image of the generic point $ \mathrm{Spec}\, k((t)) \to \mathrm{Spec}\, k[[t]] \to X$ lies in the locus where $f$ is an isomorphism, so we can lift $\mathrm{Spec}\, k((t)) \to X$ to a map $ \mathrm{Spec}\, k((t)) \to Y$, giving a diagram $\require{AMScd}$ \begin{CD} \mathrm{Spec}\, k((t)) @>>> Y\\ @V V V @VV f V\\ \mathrm{Spec}\, k[[t]] @>>> X \end{CD} Now, the valuative criterion for properness of $f$ says exactly that there is a unique lift of $\gamma$ to an arc $\mathrm{Spec}\, k[[t]] \to Y$, giving the desired bijection.

Without using the valuative criterion, the other way I can think to prove this is to recognize $f:Y\to X$ as the blowup of $X$ at some ideal sheaf, and then work locally with the natural charts on the blowup; this isn't hard, but involves involves using the nontrivial fact that all proper birational maps are blowups of some ideal.