Generic point
Encyclopedia
In mathematics
, in the fields 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.
is all of X, that is, a point that is dense in X.
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.
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 obsolete. 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 ring
s 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.)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the fields general topology
General topology
In mathematics, general topology or point-set topology is the branch of topology which studies properties of topological spaces and structures defined on them...
and particularly of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, a generic point P of a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X is an algebraic way of capturing the notion of a generic property
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
: a generic property is a property of the generic point.
Definition and motivation
A generic point of the topological space X is a point P whose closureClosure (topology)
In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
is all of X, that is, a point that is dense in X.
The terminology arises from the case of the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
of algebraic varieties. For example having a generic point is a criterion to be an irreducible set.
Examples
- For Hausdorff spaceHausdorff spaceIn topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
s, this concept is of course trivial. The only Hausdorff space that has a generic point is the singleton set - Any integral schemeGlossary of scheme theoryThis is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory...
has a (unique) generic point; in the case of an affine integral scheme (i.e., the prime spectrumSpectrum of a ringIn abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
of an integral domain) the generic point is the point associated to the prime ideal (0).
History
Points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed fieldAlgebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
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
Oscar Zariski
Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
, a colleague of Weil's at São Paulo
São Paulo
São Paulo is the largest city in Brazil, the largest city in the southern hemisphere and South America, and the world's seventh largest city by population. The metropolis is anchor to the São Paulo metropolitan area, ranked as the second-most populous metropolitan area in the Americas and among...
just after World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, 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
Kolmogorov space
In topology and related branches of mathematics, a topological space X is a T0 space or Kolmogorov space if for every pair of distinct points of X, at least one of them has an open neighborhood not containing the other. This condition, called the T0 condition, is one of the separation axioms...
and Zariski thinks in terms of the Kolmogorov quotient.)
In the rapid foundational changes of the 1950s Weil's approach became obsolete. In scheme theory, though, from 1957, generic points returned: this time à la Zariski. For example for R a discrete valuation ring
Discrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
, Spec(R) consists of two points, a generic point (coming from the prime ideal
Prime ideal
In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
{0}) and a closed point or special point coming from the unique maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
, 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
Sierpinski space
In mathematics, the Sierpiński space is a finite topological space with two points, only one of which is closed.It is the smallest example of a topological space which is neither trivial nor discrete...
of topologists. Other local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
s 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.)