ag.algebraic geometry - What are examples illustrating the usefulness of Krull (i.e., rank > 1) valuations?

Dear Pete,

This is a fairly general comment, rather than a precise answer:

General valuations were studied very deeply by Zariski, who used them as one of the
cornerstones of his investigations of birational geoemtry (including various forms
of his Main Theorem, resolution of singularities, and so on). His picture was that
different valuation rings (dominating the given local ring $mathcal O_P$ at a point
$P$ in a surface $X$, say) correspond to different germs of curves passing through the point $P$. The rank one valuations are just the germs of algebraic curves passing through $P$,
but the other valuations are like transcental curves that give additional information.

I believe that Krull himself didn't see that higher rank valuations would be related to geometry, but that it was Zariski (who studied Krull very carefully) who saw the applicability, and introduced them (and the closely related concept of normality) into
algebraic geometry. (I believe that Zariski writes about this somewhere, although I can't
remember where.)

This point of view persisted in algebraic geometry up until the Grothendieck revolution.
Maybe one of the last results proved using this view-point was the Nagata embedding theorem.

Apparently Grothendieck was very unhappy with valuation theory (and indeed tried to keep it out of Bourbaki's commutative algebra texts, without success), and the only vestige that survived in his view-point was the valuative criteria for separatedness and properness.

Let me close by using this soap-box to encourage people to read Zariski's papers. They are quite wonderful.

EDIT: I just remembered that Lang's book on algebraic geometry, which is a kind of an abridged version of Weil's Foundations, uses valuation theory at various points. (I remember that it is treated at the beginning of the book in the section dealing with foundational concepts in algebra, but I don't remember now exactly what he proves with it later on. But it is a fairly short book, so one could flip through it and see. The whole book doesn't go all that far, and so it's possible that it appears just because it was so endemic to the commutative algebra and algebraic geometry of that time period, rather than because he does anything particularly special with it.)

EDIT: After thinking about this question, I realized that the above statement about rank one valuations corresponding to germs of algebraic curves passing through a point is misleading. See my answer
here for a (hopefully) more correct discussion of the geometric intution behind various valuations.