Generic property
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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 most of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic matrix is invertible." As another example, a generic property of a space is a property that holds at most 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." (This is by Sard's theorem.)
The precise definition of "generic" depends on the context. This article discusses two examples: The case of function spaces, and the case of algebraic varieties.
[edit] 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 K.
The space Cr[M, K] is Baire space, hence any residual set is dense. This property of the function space is what makes generic property typical. 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.
[edit] 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. 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. 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.
[edit] 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, MR1288523, ISBN 978-0-471-05059-9
