Editing General position

{{Use American English|date = January 2019}}
{{Short description|Concept in algebraic geometry}}
{{refimprove|date=May 2014}}

In [[algebraic geometry]] and [[computational geometry]], '''general position''' is a notion of [[generic property|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 [[Line (geometry)|collinear]]; if three points are collinear (even stronger, if two coincide), this is a [[degeneracy (mathematics)|degenerate case]].

This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general [[theorem]]s or giving precise statements thereof, and when writing [[computer program]]s (see ''[[Generic-case complexity|generic complexity]]'').

==General linear position==
A set of points in a {{mvar|d}}-[[dimension]]al [[affine space]] ({{mvar|d}}-dimensional [[Euclidean space]] is a common example) is in '''general linear position''' (or just '''general position''') if no {{mvar|k}} of them lie in a {{math|(''k'' &minus; 2)}}-[[dimension]]al [[Flat (geometry)|flat]] for {{math|1=''k'' = 2, 3, ..., ''d'' + 1}}. These conditions contain considerable redundancy since, if the condition holds for some value {{math|''k''<sub>0</sub>}} then it also must hold for all {{mvar|k}} with {{math|2 ≤ ''k'' ≤ ''k''<sub>0</sub>}}. Thus, for a set containing at least {{math|''d'' + 1}} points in {{mvar|d}}-dimensional affine space to be in general position, it suffices that no [[hyperplane]] contains more than {{mvar|d}} points – i.e. the points do not satisfy any more linear relations than they must.<ref>{{harvnb|Yale|1968|loc=p. 164}}</ref>

A set of at most {{math|''d'' + 1}} points in general linear position is also said to be ''affinely independent'' (this is the affine analog of [[linear independence]] of vectors, or more precisely of maximal rank), and {{math|''d'' + 1}} points in general linear position in affine ''d''-space are an [[affine basis]]. See [[affine transformation]] for more.

Similarly, ''n'' vectors in an ''n''-dimensional [[vector space]] are linearly independent [[if and only if]] the points they define in [[projective space]] (of dimension {{math|''n'' &minus; 1}}) are in general linear position.

If a set of points is not in general linear position, it is called a [[degenerate case]] or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.

A fundamental application is that, in the plane, [[five points determine a conic]], as long as the points are in general linear position (no three are collinear).

== More generally ==
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. [[conic section]]s). In [[algebraic geometry]] this kind of condition is frequently encountered, in that points should impose ''independent'' conditions on curves passing through them.

For example, [[five points determine a conic]], but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.

General position is preserved under [[biregular]] maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the [[Veronese map]] is biregular; as points under the Veronese map corresponds to evaluating a degree ''d'' polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.

The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for higher degree. 

This is not sufficient, however. While nine points determine a cubic, there are configurations of nine points that are special with respect to cubics, namely the intersection of two cubics. The intersection of two cubics, which is <math>3 \times 3 = 9</math> points (by [[Bézout's theorem]]), is special in that nine points in general position are contained in a ''unique'' cubic, while if they are contained in two cubics they in fact are contained in a [[pencil (mathematics)|pencil]] (1-parameter [[linear system]]) of cubics, whose equations are the projective linear combinations of the equations for the two cubics. Thus such sets of points impose one fewer condition on cubics containing them than expected, and accordingly satisfy an additional constraint, namely the [[Cayley–Bacharach theorem]] that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.

For points in the plane or on an [[algebraic curve]], the notion of general position is made algebraically precise by the notion of a '''regular [[Divisor (algebraic geometry)|divisor]]''', and is measured by the vanishing of the higher [[sheaf cohomology]] groups of the associated [[line bundle]] (formally, [[invertible sheaf]]). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of [[intersection number]] requires sophisticated algebra. This definition generalizes in higher dimensions to hypersurfaces (codimension 1 subvarieties), rather than to sets of points, and regular divisors are contrasted with '''superabundant divisors''', as discussed in the [[Riemann–Roch theorem for surfaces]].

Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any ''k'' distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the [[cross ratio]].

== Different geometries ==
Different geometries allow different notions of geometric constraints. For example, a circle is a concept that makes sense in [[Euclidean geometry]], but not in affine linear geometry or projective geometry, where circles cannot be distinguished from ellipses, since one may squeeze a circle to an ellipse. Similarly, a parabola is a concept in affine geometry but not in projective geometry, where a parabola is simply a kind of conic. The geometry that is overwhelmingly used in algebraic geometry is projective geometry, with affine geometry finding significant but far less use.

Thus, in Euclidean geometry three non-collinear points determine a circle (as the [[circumcircle]] of the triangle they define), but four points in general do not (they do so only for [[cyclic quadrilateral]]s), so the notion of "general position with respect to circles", namely "no four points lie on a circle" makes sense. In projective geometry, by contrast, circles are not distinct from conics, and five points determine a conic, so there is no projective notion of "general position with respect to circles".

== General type ==
{{see|General type}}
General position is a property of configurations of points, or more generally other subvarieties (lines in general position, so no three concurrent, and the like). General position is an ''extrinsic'' notion, which depends on an embedding as a subvariety. Informally, subvarieties are in general position if they cannot be described more simply than others. An ''intrinsic'' analog of general position is [[general type]], and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of [[Kodaira dimension]] of a variety, and by this measure projective spaces are the most special varieties, though there are other equally special ones, meaning having negative Kodaira dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces (<math>g \geq 2</math>), and similar classifications occur in higher dimensions, notably the [[Enriques–Kodaira classification]] of [[algebraic surface]]s.

==Other contexts==
In [[intersection theory]], both in algebraic geometry and in [[geometric topology]], the analogous notion of [[Transversality (mathematics)|transversality]] is used: subvarieties in general intersect ''transversally,'' meaning with multiplicity 1, rather than being tangent or other, higher order intersections.

===General position for Delaunay triangulations in the plane===
When discussing [[Voronoi tessellation]]s and [[Delaunay triangulation]]s in the plane, a set of [[Point (geometry)|point]]s in the [[plane (mathematics)|plane]] is said to be in general position only if no four of them lie on the same circle and no three of them are collinear. The usual lifting transform that relates the Delaunay triangulation to the bottom half of a convex hull (i.e., giving each point ''p'' an extra coordinate equal to |''p''|<sup>2</sup>) shows the connection to the planar view: Four points lie on a circle or three of them are collinear exactly when their lifted counterparts are ''not'' in general linear position.

==Abstractly: configuration spaces==
In very abstract terms, '''general position''' is a discussion of [[generic property|generic properties]] of a [[configuration space (mathematics)|configuration space]]; in this context one means properties that hold on the [[generic point]] of a configuration space, or equivalently on a Zariski-open set.

This notion coincides with the [[measure theory|measure theoretic]] notion of generic, meaning [[almost everywhere]] on the configuration space, or equivalently that points chosen at random will [[almost surely]] (with probability 1) be in general position.

==Notes==
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==References==
* {{citation|first=Paul B.|last=Yale|title=Geometry and Symmetry|publisher=Holden-Day|year=1968}}

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[[Category:Algebraic geometry]]