536:, 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.
38:
547:), 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".
422:
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
577:
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
475:) 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
660:
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
426:
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
161:
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
505:
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
561:
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
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602:
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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
179:
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general
419:
polynomial at that point, this formalizes the notion that points in general position impose independent linear conditions on varieties passing through them.
520:
distinct points in the line are in general position, but projective transformations are only 3-transitive, with the invariant of 4 points being the
404:, 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.
385:
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.
669:|) 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
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maps – if image points satisfy a relation, then under a biregular map this relation may be pulled back to the original points. Significantly, the
501:). As the terminology reflects, this is significantly more technical than the intuitive geometric picture, similar to how a formal definition of
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605:
573:, and corresponds to a variety which cannot be described by simpler polynomial equations than others. This is formalized by the notion of
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For points in the plane or on an algebraic curve, the notion of general position is made algebraically precise by the notion of a
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Note that not all points in general position are projectively equivalent, which is a much stronger condition; for example, any
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532:
Different geometries allow different notions of geometric constraints. For example, a circle is a concept that makes sense in
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that any cubic that contains eight of the points necessarily contains the ninth. Analogous statements hold for higher degree.
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situation, as opposed to some more special or coincidental cases that are possible, which is referred to as
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or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.
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258:. These conditions contain considerable redundancy since, if the condition holds for some value
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meaning with multiplicity 1, rather than being tangent or other, higher order intersections.
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352:-dimensional vector space are linearly independent if and only if the points they define in
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of the triangle they define), but four points in general do not (they do so only for
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Thus, in
Euclidean geometry three non-collinear points determine a circle (as the
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is biregular; as points under the
Veronese map corresponds to evaluating a degree
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points – i.e. the points do not satisfy any more linear relations than they must.
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cubic, while if they are contained in two cubics they in fact are contained in a
377:, as long as the points are in general linear position (no three are collinear).
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this kind of condition is frequently encountered, in that points should impose
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on the configuration space, or equivalently that points chosen at random will
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408:
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172:; if three points are collinear (even stronger, if two coincide), this is a
463:), is special in that nine points in general position are contained in a
296:-dimensional affine space to be in general position, it suffices that no
604:), and similar classifications occur in higher dimensions, notably the
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366:
If a set of points is not in general linear position, it is called a
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of a configuration space, or equivalently on a
Zariski-open set.
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for a set of points, or other geometric objects. It means the
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General position for
Delaunay triangulations in the plane
693:; in this context one means properties that hold on the
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or giving precise statements thereof, and when writing
168:. Similarly, three generic points in the plane are not
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points in general linear position is also said to be
158:. Its precise meaning differs in different settings.
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of vectors, or more precisely of maximal rank), and
62:. Unsourced material may be challenged and removed.
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451:
489:, and is measured by the vanishing of the higher
373:A fundamental application is that, in the plane,
712:(with probability 1) be in general position.
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329:points in general linear position in affine
632:is used: subvarieties in general intersect
397:conditions on curves passing through them.
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122:Learn how and when to remove this message
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285:. Thus, for a set containing at least
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624:, both in algebraic geometry and in
407:General position is preserved under
60:adding citations to reliable sources
363:) are in general linear position.
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511:Riemann–Roch theorem for surfaces
677:Abstractly: configuration spaces
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700:This notion coincides with the
606:Enriques–Kodaira classification
267:then it also must hold for all
47:needs additional citations for
665:an extra coordinate equal to |
569:analog of general position is
318:(this is the affine analog of
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402:five points determine a conic
375:five points determine a conic
29:Concept in algebraic geometry
673:in general linear position.
704:notion of generic, meaning
452:{\displaystyle 3\times 3=9}
220:is a common example) is in
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628:, the analogous notion of
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493:groups of the associated
681:In very abstract terms,
477:Cayley–Bacharach theorem
652:in the plane, a set of
650:Delaunay triangulations
597:{\displaystyle g\geq 2}
222:general linear position
197:General linear position
745:Yale, Paul B. (1968),
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509:, as discussed in the
507:superabundant divisors
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140:computational geometry
747:Geometry and Symmetry
646:Voronoi tessellations
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555:Further information:
545:cyclic quadrilaterals
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339:affine transformation
201:A set of points in a
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528:Different geometries
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316:affinely independent
56:improve this article
691:configuration space
685:is a discussion of
622:intersection theory
503:intersection number
320:linear independence
300:contains more than
18:In general position
764:Algebraic geometry
687:generic properties
626:geometric topology
610:algebraic surfaces
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534:Euclidean geometry
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391:algebraic geometry
190:generic complexity
136:algebraic geometry
71:"General position"
706:almost everywhere
702:measure theoretic
575:Kodaira dimension
307:A set of at most
232:of them lie in a
185:computer programs
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16:(Redirected from
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683:general position
644:When discussing
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499:invertible sheaf
491:sheaf cohomology
461:BĂ©zout's theorem
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381:More generally
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356:(of dimension
348:vectors in an
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252:= 2, 3, ...,
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67:Find sources:
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45:This article
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749:, Holden-Day
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54:Please help
49:verification
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522:cross ratio
497:(formally,
495:line bundle
459:points (by
395:independent
344:Similarly,
242:dimensional
207:dimensional
739:References
341:for more.
298:hyperplane
292:points in
238:− 2)
148:genericity
82:newspapers
729:Yale 1968
589:≥
567:intrinsic
563:extrinsic
438:×
409:biregular
361:− 1
224:(or just
170:collinear
758:Category
731:, p. 164
484:regular
228:) if no
181:theorems
112:May 2014
656:in the
486:divisor
96:scholar
654:points
469:pencil
465:unique
389:). In
337:. See
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91:
84:
77:
69:
716:Notes
689:of a
658:plane
271:with
187:(see
103:JSTOR
89:books
648:and
273:2 ≤
247:for
245:flat
138:and
75:news
671:not
620:In
608:of
327:+ 1
312:+ 1
290:+ 1
256:+ 1
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134:In
58:by
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277:≤
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667:p
663:p
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282:0
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