General position

In algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.

For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case.

This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see generic complexity).

A set of at least ${\displaystyle d+1}$ points in ${\displaystyle d}$-dimensional affine space (${\displaystyle d}$-dimensional Euclidean space is a common example) is said to be in general linear position (or just general position) if no hyperplane contains more than ${\displaystyle d}$ points — i.e. the points do not satisfy any more linear relations than they must. In more generality, a set containing ${\displaystyle k}$ points, for arbitrary ${\displaystyle k}$, is in general linear position if and only if no ${\displaystyle (k-2)}$-dimensional flat contains all ${\displaystyle k}$ points.

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