Why should the normalization of a ring correspond to the intersection of valuation rings containing it? I am looking for a geometric explanation, if possible.
I understand that normalization at a point of a variety corresponds to making sure every bounded function in a neighborhood of that point extends over the point. But I'm having trouble translating that statement into the algebraic one.
There is some discussion of this issue by Kollár on page 20 here, but don't see how "definition" 1.23 connects to the preceding discussion. And from there, it seems like a stretch to get to the correct definition (using inclusions instead of arbitrary homomorphisms, and using arbitrary valuation rings instead of just DVRs).
Edit: As pointed out in the comments, any Noetherian valuation ring is a DVR (or field). In geometric applications, everything is Noetherian, so that helps a bit. I would still appreciate an answer explaining Kollár's remarks about the geometric content of the DVR viewpoint on normalization a bit more.