I am familiar with basic algebraic geometry in the tradition of Hartshorne's book. Discrete valuation rings appear there in the criteria for separatedness/properness, and are used to define the order of a pole/zero of a meromorphic function on a nonsingular variety. Furthermore, points of a nonsingular projective curve are in bijection with valuation rings on of the function field.
In Ravi Vakil's notes on algebraic geometry there is a quote of a letter from Serre to Grothendieck. It appears that Grothendieck disliked valuation rings, whereas Weil thought that valuation rings should play a central role. I have heard from other sources that valued fields have applications in geometry.
My questions are:
1) What is the importance of valuation rings in algebraic geometry other than what I mentioned in the first paragraph? For example, does the topology on a valuation ring play a role? Are valuation rings which are non-discrete important? Can they be used in intersection theory?
2) Is there any old paper, accessible as much as possible to someone who is familiar with the language of schemes, where valuation rings / valued fields are used?
Best Answer
Tropical algebraic geometry uses valuations on fields a lot. If you have a variety over a field with a non-archimedean valuation, you can consider the non-archimedean amoeba (set of all component-wise valuations), which is a polyhedral complex, whose structure can be used to study the variety (including its intersection theory in some cases). I don't know a good paper on the top of my head, but The tropical Grassmannian by Speyer and Sturmfels should do as introduction.