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 square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If f : M → N is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f."
There are many different notions of "generic" in mathematics, with corresponding dual notions of "almost none" ; the two main classes are:
There are several natural examples where those notions are not equal. For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero.

In measure theory

In measure theory, a generic property is one that holds almost everywhere. The dual concept is a null set, that is, a set of measure zero.

In probability

In probability, a generic property is an event that occurs almost surely, meaning that it occurs with probability 1. For example, the law of large numbers states that the sample mean converges almost surely to the population mean. This is the definition in the measure theory case specialized to a probability space.

In discrete mathematics

In discrete mathematics, one uses the term almost all to mean cofinite, cocountable, for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graphs.

In topology

In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a closed nowhere dense set, or more generally a meagre set.
However, density alone is not sufficient to characterize a generic property. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. Since it does not make sense to say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical. Consequently, we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not.
For applications, if a property holds on a residual set, it may not hold for every point, but perturbing it slightly will generally land one inside the residual set, and these are thus the most important case to address in theorems and algorithms.

In function spaces

A property is generic in Cr if the set holding this property contains a residual subset in the Cr topology. Here Cr is the function space whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N.
The space Cr, of Cr mappings between M and N, is a Baire space, hence any residual set is dense. This property of the function space is what makes generic properties typical.

In algebraic geometry

Algebraic varieties

A property of an irreducible algebraic variety X is said to be true generically if it holds except on a proper Zariski-closed subset of X, in other words, if it holds on a non-empty Zariski-open subset. This definition agrees with the topological one above, because for irreducible algebraic varieties any non-empty open set is dense.
For example, by the Jacobian criterion for regularity, a generic point of a variety over a field of characteristic zero is smooth. This is true because the Jacobian criterion can be used to find equations for the points which are not smooth: They are exactly the points where the Jacobian matrix of a point of X does not have full rank. In characteristic zero, these equations are non-trivial, so they cannot be true for every point in the variety. Consequently, the set of all non-regular points of X is a proper Zariski-closed subset of X.
Here is another example. Let f : X → Y be a regular map between two algebraic varieties. For every point y of Y, consider the dimension of the fiber of f over y, that is, dim f−1. Generically, this number is constant. It is not necessarily constant everywhere. If, say, X is the blowup of Y at a point and f is the natural projection, then the relative dimension of f is zero except at the point which is blown up, where it is dim Y - 1.
Some properties are said to hold very generically. Frequently this means that the ground field is uncountable and that the property is true except on a countable union of proper Zariski-closed subsets. For instance, this notion of very generic occurs when considering rational connectedness. However, other definitions of very generic can and do occur in other contexts.

Generic point

In algebraic geometry, a generic point of an algebraic variety is a point whose coordinates do not satisfy any other algebraic relation than those satisfied by every point of the variety. For example, a generic point of an affine space over a field is a point whose coordinates are algebraically independent over.
In scheme theory, where the points are the sub varieties, a generic point of a variety is a point whose closure for the Zariski topology is the whole variety.
A generic property is a property of the generic point. For any reasonable property, it turns out that the property is true generically on the subvariety if and only if the property is true at the generic point. Such results are frequently proved using the methods of limits of affine schemes developed in EGA IV 8.

General position

A related concept in algebraic geometry is general position, whose precise meaning depends on the context. For example, in the Euclidean plane, three points in general position are not collinear. This is because the property of not being collinear is a generic property of the configuration space of three points in R2.

Genericity results