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Find link is a tool written by Edward Betts.Longer titles found: Projective line over a ring (view), Projective linear group (view), Real projective line (view)
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alternate case: projective line
Dual number
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"cyclic rotation" of the dual number plane occurs as a motion of its projective line. According to Isaak Yaglom,: 92–93 the cycle Z = {z : y = αx2} isRuled surface (2,901 words) [view diff] exact match in snippet view article find links to article
in which case "straight line" is understood to mean an affine or projective line. A surface in 3-dimensional Euclidean space is called a ruled surfaceDeligne–Mumford stack (604 words) [view diff] no match in snippet view article find links to article
In algebraic geometry, a Deligne–Mumford stack is a stack F such that the diagonal morphism F → F × F {\displaystyle F\to F\times F} is representable,Algebraic manifold (201 words) [view diff] exact match in snippet view article find links to article
one example of a complex algebraic manifold, since it is the complex projective line. Elliptic curves Grassmannian Algebraic geometry and analytic geometryGromov's inequality for complex projective space (438 words) [view diff] exact match in snippet view article find links to article
the areas of rational 2-cycles representing the class of the complex projective line C P 1 ⊂ C P n {\displaystyle \mathbb {CP} ^{1}\subset \mathbb {CP}Real projective plane (2,704 words) [view diff] no match in snippet view article find links to article
origin; then the parallel plane which does pass through the origin (a projective "line") is called the line at infinity. (See § Homogeneous coordinates belowGluing schemes (787 words) [view diff] exact match in snippet view article find links to article
t^{-1}\leftrightarrow u} , then the resulting scheme is, at least visually, the projective line P 1 {\displaystyle \mathbb {P} ^{1}} . The category of schemes admitsJumping line (519 words) [view diff] exact match in snippet view article find links to article
or exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behaviorWeil conjectures (7,942 words) [view diff] exact match in snippet view article find links to article
where U is the points the projective line with non-singular fibers, and j is the inclusion of U into the projective line, and E is the sheaf with fibersMathieu group M24 (3,022 words) [view diff] exact match in snippet view article find links to article
the subgroups fix a point, duad, octad, duum, sextet, triad, trio, projective line, or octern, as described below. Todd (1966) gave the character tablesVanishing cycle (456 words) [view diff] exact match in snippet view article find links to article
example, in a map from a connected complex surface to the complex projective line, a generic fiber is a smooth Riemann surface of some fixed genus gComplex projective plane (527 words) [view diff] exact match in snippet view article find links to article
dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groupsRuled variety (836 words) [view diff] exact match in snippet view article find links to article
{\displaystyle k} is ruled if it is birational to the product of the projective line with some variety over k {\displaystyle k} . A variety is uniruledDegeneration (algebraic geometry) (596 words) [view diff] exact match in snippet view article
variety (or a scheme) to a curve C with origin 0 (e.g., affine or projective line), the fibers π − 1 ( t ) {\displaystyle \pi ^{-1}(t)} form a familyBloch sphere (3,793 words) [view diff] exact match in snippet view article find links to article
two-dimensional Hilbert space, the space of all such states is the complex projective line C P 1 . {\displaystyle \mathbb {C} \mathbf {P} ^{1}.} This is the BlochBirkhoff factorization (1,008 words) [view diff] exact match in snippet view article find links to article
Birkhoff–Grothendieck theorem of Grothendieck (1957) that vector bundles over the projective line are sums of line bundles. There are several variations where the generalArithmetic dynamics (1,668 words) [view diff] exact match in snippet view article find links to article
Baker, Matthew (2010). Potential theory and dynamics on the Berkovich projective line. Mathematical Surveys and Monographs. Vol. 159. Providence, RI: AmericanOriented projective geometry (832 words) [view diff] exact match in snippet view article find links to article
n {\displaystyle \mathbb {R} ^{n}} excluding the origin. Oriented projective line, T 1 {\displaystyle \mathbb {T} ^{1}} : ( x , w ) ∈ R ∗ 2 {\displaystyleRobert Rumely (375 words) [view diff] case mismatch in snippet view article find links to article
Society 145, 2000) Potential Theory and Dynamics on the Berkovich Projective Line (Mathematical Surveys and Monographs 159, 2010) Capacity Theory withArtin–Schreier curve (1,090 words) [view diff] exact match in snippet view article find links to article
branched covering C → P 1 {\displaystyle C\to \mathbb {P} ^{1}} of the projective line of degree p {\displaystyle p} . Such a cover is necessarily cyclicProjective frame (785 words) [view diff] exact match in snippet view article find links to article
coordinates of h(a) on the canonical frame of Pn(K). In the case of a projective line, a frame consists of three distinct points. If P1(K) is identifiedRiemann–Hilbert correspondence (1,331 words) [view diff] exact match in snippet view article find links to article
number. This equation has regular singularities at 0 and ∞ in the projective line P1. The local solutions of the equation are of the form cza for constantsGlossary of classical algebraic geometry (11,193 words) [view diff] exact match in snippet view article find links to article
itself cross-ratio The cross-ratio is an invariant of 4 points on a projective line. crunode Crunode is an archaic term for a node, a double point withHopf surface (866 words) [view diff] exact match in snippet view article find links to article
the projective line if α m = β n {\displaystyle \alpha ^{m}=\beta ^{n}} for some positive integers m and n, with the map to the projective line givenGauss–Manin connection (1,100 words) [view diff] exact match in snippet view article find links to article
free parameter describing the curve; it is an element of the complex projective line (the family of hypersurfaces in n − 1 {\displaystyle n-1} dimensionsZassenhaus group (334 words) [view diff] exact match in snippet view article find links to article
linear group PSL2(Fq) for q > 3 odd, acting on the q + 1 points of the projective line. It has order (q + 1)q(q − 1)/2. The projective general linear groupBerkovich space (1,582 words) [view diff] exact match in snippet view article find links to article
{\displaystyle k} as a dense subspace. One can also define the Berkovich projective line P 1 {\displaystyle \mathbb {P} ^{1}} by adjoining to A 1 {\displaystyleDerived noncommutative algebraic geometry (4,702 words) [view diff] no match in snippet view article find links to article
In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derivedModular lambda function (3,503 words) [view diff] exact match in snippet view article find links to article
cross ratio of the branch points of a ramified double cover of the projective line by the elliptic curve C / ⟨ 1 , τ ⟩ {\displaystyle \mathbb {C} /\langleW-curve (261 words) [view diff] exact match in snippet view article find links to article
a line. A 1-dimensional W-curve (read: the motion of a point on a projective line) is determined by such a series. The German "W-Kurve" sounds almostConfiguration space (physics) (1,410 words) [view diff] exact match in snippet view article
in C P 1 {\displaystyle \mathbb {C} \mathbf {P} ^{1}} , the complex projective line, also known as the Bloch sphere. It is complex, because a quantum-mechanicalProjective Hilbert space (826 words) [view diff] exact match in snippet view article find links to article
complex Hilbert space (the space describing one qubit) is the complex projective line C P 1 {\displaystyle \mathbb {C} \mathbf {P} ^{1}} . This is knownMotive (algebraic geometry) (4,886 words) [view diff] exact match in snippet view article
and crystalline cohomology. The general hope is that equations like [projective line] = [line] + [point] [projective plane] = [plane] + [line] + [point]Veronese surface (941 words) [view diff] exact match in snippet view article find links to article
{\displaystyle n=1,d=1} the Veronese map is simply the identity map on the projective line. For n = 1 , d = 2 , {\displaystyle n=1,d=2,} the Veronese varietyUniform matroid (1,001 words) [view diff] exact match in snippet view article find links to article
n − 1 {\displaystyle n-1} or more elements (because otherwise the projective line over that field would have fewer than n {\displaystyle n} points):Splitting principle (626 words) [view diff] exact match in snippet view article find links to article
splitting principle for holomorphic vector bundles on the complex projective line H. Blane Lawson and Marie-Louise Michelsohn, Spin Geometry, PropositionMinimal model program (1,353 words) [view diff] exact match in snippet view article find links to article
unique, though there is a unique one isomorphic to the product of the projective line and a curve. A somewhat subtle point is that even though a surfaceGeneral position (1,469 words) [view diff] exact match in snippet view article find links to article
dimension. For algebraic curves, the resulting classification is: projective line, torus, higher genus surfaces ( g ≥ 2 {\displaystyle g\geq 2} ), andPlane at infinity (976 words) [view diff] exact match in snippet view article find links to article
of parallel planes in affine 3-space will intersect each other in a projective line (a line at infinity) in the plane at infinity. Also, every plane inIntegral polytope (947 words) [view diff] exact match in snippet view article find links to article
the Segre embedding of the n {\displaystyle n} -fold product of the projective line.[citation needed] In algebraic geometry, an important instance of latticePicard–Lefschetz theory (795 words) [view diff] exact match in snippet view article find links to article
{\displaystyle (k+1)} -dimensional projective complex manifold to the projective line P1. Also suppose that all critical points are non-degenerate and lieFinite geometry (2,841 words) [view diff] exact match in snippet view article find links to article
1 (exactly one line): All points lie on the unique line, called a projective line. Dimension 2: There are at least 2 lines, and any two lines meet. ABranched covering (1,711 words) [view diff] exact match in snippet view article find links to article
branched covering of the corresponding projective elliptic curve to the projective line. The previous example may be generalized to any algebraic plane curveBorel–Weil–Bott theorem (1,898 words) [view diff] exact match in snippet view article find links to article
} The flag variety G/B may be identified with the complex projective line CP1 with homogeneous coordinates X, Y and the space of the global sectionsWigner's theorem (4,735 words) [view diff] exact match in snippet view article find links to article
{\underline {\Psi }}_{2}} (i.e. linearly independent lines) there is a projective line of rays of the form λ 1 Ψ 1 + λ 2 Ψ 2 _ {\displaystyle {\underlineRank 3 permutation group (765 words) [view diff] exact match in snippet view article find links to article
6-point permutation representation; two classes A9 L2(8):3 120 = 1+56+63 Projective line P1(8); two classes A10 (A5×A5):4 126 = 1+25+100 Sets of 2 blocks ofKarl Georg Christian von Staudt (1,614 words) [view diff] exact match in snippet view article find links to article
Giovanni Vailati who studied the circular order property of the real projective line. The science of this order requires a quaternary relation called point-pairCanonical bundle (2,548 words) [view diff] exact match in snippet view article find links to article
map is given by homogeneous coordinates [1: x] as a morphism to the projective line. The rational normal curve for higher genus hyperelliptic curves arisesMathieu group (2,168 words) [view diff] exact match in snippet view article find links to article
Zassenhaus groups notably include the projective general linear group of a projective line over a finite field, PGL(2,Fq), which is sharply 3-transitive (seePin group (2,502 words) [view diff] exact match in snippet view article find links to article
"the projectivization of a 2n-gon in the circle is an n-gon in the projective line". In 3 dimensions the situation is as follows. The Clifford algebraKummer surface (1,547 words) [view diff] exact match in snippet view article find links to article
ordinary double point p, near which K looks like a quadratic cone. Any projective line through p then meets K with multiplicity two at p, and will thereforeČech cohomology (3,378 words) [view diff] exact match in snippet view article find links to article
coherent sheaf cohomology of Ω 1 {\displaystyle \Omega ^{1}} on the projective line P C 1 {\displaystyle \mathbb {P} _{\mathbb {C} }^{1}} using the ČechProjective polyhedron (2,111 words) [view diff] exact match in snippet view article find links to article
The projectivization of a 2r-gon (in the circle) is an r-gon (in the projective line), and accordingly the quotient groups, subgroups of PO(2) and PSO(2)Special unitary group (5,722 words) [view diff] exact match in snippet view article find links to article
hyperbolic plane geometry. Indeed, for a point [z, 1] in the complex projective line, the action of SU(1,1) is given by ( u v v ∗ u ∗ ) [ z , 1 ] = [ uAscher Wagner (687 words) [view diff] exact match in snippet view article find links to article
D.; Siemons, Johannes; Wagner, Ascher (1986). "Regular sets on the projective line" (PDF). Journal of Geometry. 27 (2): 188–194. doi:10.1007/BF01224556Inversive geometry (4,386 words) [view diff] exact match in snippet view article find links to article
reciprocation is the apparent operation, this procedure leads to the complex projective line, often called the Riemann sphere. It was subspaces and subgroups ofElliptic surface (1,883 words) [view diff] exact match in snippet view article find links to article
Enriques surface is elliptic, and has an elliptic fibration over the projective line. Kodaira surfaces Dolgachev surfaces Shioda modular surfaces Most ofComplex projective space (3,929 words) [view diff] exact match in snippet view article find links to article
The Riemann sphere, the one-dimensional complex projective space, i.e. the complex projective line.Complex geometry (3,677 words) [view diff] exact match in snippet view article find links to article
A typical example of a complex space is the complex projective line. It may be viewed either as the sphere, a smooth manifold arising from differentialAffine space (7,537 words) [view diff] exact match in snippet view article find links to article
For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane)Combinatorial species (2,907 words) [view diff] exact match in snippet view article find links to article
(coordinatizated by a field) with the infinite point and obtaining a projective line. There are a variety of other manipulations which may be performedSpin-weighted spherical harmonics (2,161 words) [view diff] exact match in snippet view article find links to article
antiholomorphic vector bundle O(2s) of the Serre twist on the complex projective line CP1. A section of the latter bundle is a function g on C2\{0} satisfyingSuperelliptic curve (1,413 words) [view diff] exact match in snippet view article find links to article
branched covering C → P 1 {\displaystyle C\to \mathbb {P} ^{1}} of the projective line of degree m ≥ 2 {\displaystyle m\geq 2} coprime to the characteristicKobayashi metric (2,246 words) [view diff] exact match in snippet view article find links to article
which the Kobayashi pseudometric is identically zero: the complex projective line CP1 or more generally complex projective space CPn, C−{0} (using theDuality (projective geometry) (5,688 words) [view diff] exact match in snippet view article
a pencil of hyperplanes in higher dimensions. A line segment on a projective line has as its dual the shape swept out by these lines or hyperplanes,Complex dynamics (4,690 words) [view diff] exact match in snippet view article find links to article
C to itself. It is helpful to view this as a map from the complex projective line C P 1 {\displaystyle \mathbf {CP} ^{1}} to itself, by adding a pointInvariant of a binary form (2,706 words) [view diff] exact match in snippet view article find links to article
because such a curve can be represented as a double cover of the projective line branched at 6 points, and the 6 points can be taken as the roots ofFubini–Study metric (5,285 words) [view diff] exact match in snippet view article find links to article
defines polar coordinate one-forms on the 4-sphere (the quaternionic projective line) as r d r = + x d x + y d y + z d z + t d t r 2 σ 1 = − t d x − z dQuartic function (6,853 words) [view diff] exact match in snippet view article find links to article
pencil is given by the forms λF1 + μF2 for any point [λ, μ] in the projective line — in other words, where λ and μ are not both zero, and multiplyingField with one element (3,811 words) [view diff] exact match in snippet view article find links to article
over F1, introducing extensions of F1 and using them to handle the projective line P1 over F1. Algebraic numbers were treated as maps to this P1, andDegenerate conic (2,645 words) [view diff] exact match in snippet view article find links to article
then loops around to a > 1 , {\displaystyle a>1,} since pencils are a projective line. Note that this parametrization has a symmetry, where inverting theChern–Weil homomorphism (2,782 words) [view diff] exact match in snippet view article find links to article
normalization property, one computes the first Chern class of the complex projective line; see Chern class#Example: the complex tangent bundle of the RiemannUnum (number format) (2,877 words) [view diff] exact match in snippet view article
finite set of points between one and infinity, to quantify the entire projective line except for four points: the two exceptions, 0 and ∞, and then 1 andDivisor (algebraic geometry) (6,612 words) [view diff] exact match in snippet view article
projective curve X, the degree gives a homomorphism deg: Cl(X) → Z. For the projective line P1 over a field k, the degree gives an isomorphism Cl(P1) ≅ Z. ForEnriques–Kodaira classification (4,245 words) [view diff] exact match in snippet view article find links to article
this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface X is calledIsomonodromic deformation (2,928 words) [view diff] exact match in snippet view article find links to article
_{i=1}^{n}{\frac {A_{i}}{x-\lambda _{i}}}y} where x takes values in the complex projective line C P 1 {\displaystyle \mathbb {CP} ^{1}} , the y takes values in C nMain theorem of elimination theory (1,567 words) [view diff] exact match in snippet view article find links to article
an algebraic variety. If one extends L y {\displaystyle L_{y}} to a projective line P y , {\displaystyle P_{y},} the equation of the projective completionGauge theory (mathematics) (11,468 words) [view diff] exact match in snippet view article
equivalent to rational maps of degree k {\displaystyle k} from the complex projective line C P 1 {\displaystyle \mathbb {CP} ^{1}} to itself, where k {\displaystyleGlossary of algebraic geometry (12,496 words) [view diff] exact match in snippet view article find links to article
{\displaystyle g=0} . rational curves, i.e. the curve is birational to the projective line P 1 {\displaystyle \mathbb {P} ^{1}} . (b) g = 1 {\displaystyle g=1}Morphism of schemes (5,034 words) [view diff] exact match in snippet view article find links to article
from X to the affine line A 1 {\displaystyle \mathbb {A} ^{1}} or the projective line P 1 . {\displaystyle \mathbb {P} ^{1}.} A rational map is dominantArrangement of lines (6,722 words) [view diff] exact match in snippet view article find links to article
replaced in the projective plane by single cells that are crossed by the projective line at infinity. Due to projective duality, many statements about the combinatorialK-stability of Fano varieties (9,391 words) [view diff] exact match in snippet view article find links to article
{\mathcal {L}})\to \mathbb {C} } to a test configuration over the complex projective line P 1 = C ∪ { ∞ } {\displaystyle \mathbb {P} ^{1}=\mathbb {C} \cup \{\inftyGlossary of invariant theory (4,629 words) [view diff] exact match in snippet view article find links to article
group. cross ratio The cross ratio is an invariant of 4 points of a projective line. cubic (Adjective) Degree 3 (Noun) A form of degree 3 cubicovariantQuadric (algebraic geometry) (3,537 words) [view diff] exact match in snippet view article
^{2}} is called a conic. A split conic over k is isomorphic to the projective line P 1 {\displaystyle \mathbf {P} ^{1}} over k, embedded in P 2 {\displaystyleTimeline of manifolds (1,178 words) [view diff] exact match in snippet view article find links to article
presented as ramified covering spaces of the Riemann sphere (the complex projective line). 1854 Bernhard Riemann Riemannian metrics give an idea of intrinsicFour-dimensional Chern–Simons theory (1,480 words) [view diff] exact match in snippet view article find links to article
{\displaystyle C} is C P 1 {\displaystyle \mathbb {CP} ^{1}} the complex projective line. The form ω {\displaystyle \omega } has two poles; either a single