Generic property

Definitions: measure theory

In measure theory, a generic property is one that holds almost everywhere, meaning "with probability 1", with the dual concept being null set, meaning "with probability 0".

Probability

In probability, one speaks of trials instead of a space, so one instead says that a property holds almost surely for "with probability 1". For example, the Law of large numbers states that the sample mean converges almost surely to the population mean.

Discrete mathematics

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

Definitions: 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 (a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set, or more generally a meagre set (a countable union of nowhere dense closed sets).

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 (by nowhere density of the components of the meagre set), and these are thus the most important case to address in theorems and algorithms.

Function spaces

A property is generic in C^r if the set holding this property contains a residual subset in the C^r topology. Here C^r is the function space whose members are continuous functions with r continuous derivatives from a manifold M to a manifold N.

The space C^r(M, N), of C^r 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.

Algebraic geometry

Algebraic varieties

A property of an 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. By the definition of algebraic varieties, they are irreducible spaces, hence 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 statement is known as generic smoothness.) 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(y). 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 (i.e., the property holds on a dense G_δ set). 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

Main article: Generic point

In scheme theory, one formalizes the notion of a generic property by adding additional points for each subvariety, called the "generic point" of the subvariety. Then 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 (in the sense of being true on an open dense subset) 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

Main article: General position

A related concept in algebraic geometry is general position – points are in general position if they satisfy no more equations than are necessary. For instance, no 3 points are collinear. This is a notion of a generic property of a configuration space.

Genericity results

Sard's theorem: If $f\colon M \to N$ is a smooth function between smooth manifolds, then a generic point of N is not a critical value of f – critical values of f are a null set in N.
Jacobian criterion / generic smoothness: A generic point of a variety over a field of characteristic zero is smooth.

References

Wiggins, Stephen (2003), Introduction to applied nonlinear dynamical systems and chaos, Berlin, New York: Springer-Verlag, ISBN 978-0-387-00177-7
Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9, MR 1288523