Generic point

In mathematics, in the fields of general topology and particularly of algebraic geometry, a generic point "P" of a topological space "X" is an algebraic way of capturing the notion of a generic property: a generic property is a property of "the" generic point.

Definition

Formally, a generic point is a point P such that every point "Q" of "X" is a "specialization" of "P", in the sense of the specialization order (or preorder): the closure of "P" is the entire set: it is dense.

This concept is only non-trivial for spaces that are not Hausdorff spaces, because a Hausdorff space with a generic point "P" can only be the singleton set {"P"}. The terminology arises from the case of the Zariski topology of algebraic varieties. For example having a generic point is a criterion to be an irreducible set.

History

In the foundational approach of André Weil, developed in his "Foundations of Algebraic Geometry", generic points played an important role, but were handled in a different manner. For an algebraic variety "V" over a field "K", "generic" points of "V" were a whole class of points of "V" taking values in a universal domain Ω, an algebraically closed field containing "K" but also an infinite supply of fresh indeterminates. This approach worked, without any need to deal directly with the topology of "V" ("K"-Zariski topology, that is), because the specializations could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s).

This was at a cost of there being a huge collection of equally-generic points. Oscar Zariski, a colleague of Weil's at São Paulo just after World War II, always insisted that generic points should be unique. (This can be put back into topologists' terms: Weil's idea fails to give a Kolmogorov space and Zariski thinks in terms of the Kolmogorov quotient.)

In the rapid foundational changes of the 1950s Weil's approach became obsolescent. In scheme theory, though, from 1957, generic points returned: this time "à la Zariski". For example for "R" a discrete valuation ring, "Spec"("R") consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal, For morphisms to "Spec"("R"), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point. Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists. Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions. The discrete valuation case is much like the complex unit disk, for these purposes.)