I recently came across a paper using a "universal domain" to discuss "generic points" of a variety, using Weil's foundations of algebraic geometry instead of Grothendieck's. First I had to learn that stuff, then I wanted to translate it. This lead to a more systematic study of what it means to be a point of a variety or scheme, in the various different definitions.
So in this post I will explain closed points, generic points, points in general position, schematic points, generalized points, rational points, geometric points, and in particular, which of these notions can be considered a particular case of another of these. I will try to give you a hint why one wants to generalize the ordinary (closed) points of a variety that much, to answer the question in the title: "What's the point of this?".
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