generic point

When mathematicians say 'a generic line through two points' or 'a generic polynomial', they mean one with no special accidental coincidences. Scheme theory makes this folklore literal: it adds an actual point to the space, the generic point, that sits at no special location and whose properties are exactly the properties shared by almost all ordinary points.

Formally, a point of a topological space whose closure is the entire space is called a generic point of that space. An irreducible scheme has a unique generic point. In the spectrum Spec R of an integral domain it is the zero ideal (0), the unique minimal prime; its closure is everything, and its local ring is the function field of the variety. More generally each irreducible closed subset of a scheme has its own unique generic point, the minimal prime of that subset.

Generic points are the device by which a single point carries the behavior of a whole open dense set. A property that holds at the generic point holds on a nonempty open neighborhood — 'generically' — which is the rigorous content of the classical phrase. This is why scheme-theoretic statements can replace cumbersome 'for all but finitely many' arguments with a clean assertion about one well-chosen point.