
	Where results make sense

Topic: Complex projective line

Related Topics

Riemann sphere

Complex projective space

	More projective groups (Site not responding. Last check: 2007-10-18)
		Complex variable theory shows that these mappings are conformal (angle preserving) transformations at all points.
		As in the last example, we may define the cross-ratio of four points (a complex number this time) and this is preserved by these transformations.
		As in the projective line case, one may take standard reference points to be: [1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 1].
	www-history.mcs.st-and.ac.uk /~john/MT4521/Lectures/L19.html (228 words)

	Projective line - Wikipedia, the free encyclopedia
		The projective line may also be thought of as the line K together with an idealised point at infinity.
		It is in constant use in complex analysis, algebraic geometry and complex manifold theory, as the simplest example of a compact Riemann surface.
		The projective line is a fundamental example of an algebraic curve.
	en.wikipedia.org /wiki/Projective_line (991 words)

	[No title]
		Even though C is investigating the geometry of a line, since it is the compl\hich\af4\dbch\af4\loch\f4 ex projective line, she can consider points, segments, angles, triangles, circles, etc. on the line and, most importantly, use properties of the complex numbers like taking conjugates, to study and prove results about various geometric figures.
		Indeed, C\hich\f4 \rquote \loch\f4 s complex projective line can be thought as a plane with a point \hich\af4\dbch\af4\loch\f4 a\hich\af4\dbch\af4\loch\f4 t infinity, although it never occurred to her before to think of it that way.
		He realizes that certain lines on the sphere are sent to circles with an infinite radius, namely\hich\af4\dbch\af4\loch\f4 \hich\f4 those lines passing through the north pole and that her geometry which seemed to be essentially bounded is not intrinsically so.
	www.math.mcgill.ca /rags/seminar/JPM-Chapitre1.rtf (3405 words)

	[No title]
		A (complex) affine or projective curve is an closed algebraic variety over $\C$ of dimension 1 either in affine or projective space.
		The complex structure of the quotient is the complex structure induced from $\C$.
		As explained in Section 2 in both cases the complex projective line $\P^1$ is the desingularization.
	www.ma.utexas.edu /mp_arc/papers/92-12 (6353 words)

	Projective spaces
		These ideas were developed into the notion of projective geometry which was studied in the 17th century by (among other) Girard Desargues (1591 to 1661).
		A line from P parallel to the line m does not meet m and so no point on m corresponds to the point x on l.
		Thus a projective line is really rather like circle and when we add a line at infinity to the plane this is more like a circle at infinity.
	www-groups.dcs.st-and.ac.uk /~john/MT4521/Lectures/L16.html (460 words)

	Complex projective space - Wikipedia, the free encyclopedia
		The case n = 1 gives the Riemann sphere (also called the complex projective line), and the case n = 2 the complex projective plane.
		Complex projective space is a complex manifold that may be described by n+1 complex coordinates as
		It is BU(1), the classifying space of U(1), in the sense of homotopy theory, and so classifies complex line bundles; equivalently it accounts for the first Chern class.
	en.wikipedia.org /wiki/Complex_projective_space (375 words)

	Research/Chain Geometry
		The aim of the project is to investigate various aspects of chain geometry, in particular the (generalized) chain geometries arising from the projective line P(R) over a ring R containing a field F which is not necessarily commutative.
		The projective line over this ring is represented by lines of a special linear complex, i.e., the set of all lines in a 3-dimensional projective space that meet a fixed axis; only this axis represents no point of the projective line over the ternions.
		Havlicek: On Distant-Isomorphisms of Projective Lines, Aequationes Mathematicae 69 (2005), 146-163.
	www.geometrie.tuwien.ac.at /havlicek/proj303.html (1607 words)

	Midwest Algebraic Geometry Conference
		We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane, which is itself a deformation of the usual cohomology ring.
		We prove a criterion for the existence of a vector bundle on projective n-space by the existence of certain vector bundles in (n-1)-space.
		X is projectively normal, ideal-theoretically defined by quadrics, and all of the syzygies are generated by linear ones.
	www.nd.edu /~rosen/MAGC97/magc97/magc97.html (7511 words)

	[No title]
		This is the set of complex numbers together with a single point at infinity.
		The natural operations on the projective line are the linear fractional transformations (LFT for short) x goes to (ax+b)/(cx+d) where ad-bc is non zero.
		In other words, given 3 distinct points x,y,z in the projective line, and three other distinct points x',y',z', there exists a unique LFT sending x to x', y to y', z to z'.
	www.math.niu.edu /~rusin/known-math/95/dessins (2124 words)

	Octonionic Projective Geometry
		Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters.
		We have already met one example of a projective plane in Section 2.1: the smallest one of all, the Fano plane.
		The dimension of a projective space is defined to be one less than the minimal cardinality of a set that spans the whole space.
	math.ucr.edu /home/baez/octonions/node8.html (1306 words)

	Introduction to Complex Manifolds (Site not responding. Last check: 2007-10-18)
		Complex manifolds play a great role in many areas of modern mathematics.
		Complex manifold are of importance also in the transcendental approach to algebraic geometry in higher dimension; a pioneer here was Hodge.
		A basic question is then to find conditions for a complex manifold to admit imbedding in a complex projective space of sufficiently high dimension -- an answer to this question is provided by the Kodaira imbedding theorem.
	www.maths.lth.se /matematiklu/personal/jaak/Complex-Manifolds.html (218 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-18)
		Elements (points, straight lines, planes, etc.), generated by extending a given affine space to a compact space.
		, which is homeomorphic to the complex projective straight line or the Riemann sphere
		For instance, to parallel straight lines, in the projective plane
	eom.springer.de /i/i050900.htm (292 words)

	PlanetMath: Hopf bundle (Site not responding. Last check: 2007-10-18)
		, the complex projective line by the natural projection.
		Cross-references: generator, restriction, homeomorphic, projection, complex projective line, map, structure
		The most remarkable fact (for Geometry) is that we can give the coordinates for our calculation to this Bundle as the example of Principal Fibre Bundle.
	planetmath.org /encyclopedia/HopfFibration.html (104 words)

	Oxford University Press (Site not responding. Last check: 2007-10-18)
		Complex hyperbolic geometry is a particularly rich area of study, enhanced by the confluence of several areas of research including Riemannian geometry, complex analysis, symplectic and contact geometry, Lie group theory, and harmonic analysis.
		The boundary of complex hyperbolic geometry, known as spherical CR or Heisenberg geometry, is equally rich, and although there exist accounts of analysis in such spaces there is currently no account of their geometry.
		Motivated by applications of the theory to geometric structures, moduli spaces and discrete groups, it is designed to provide an introduction to this fascinating and important area and invite further research and development.
	www.oup.com /ca/isbn/0-19-853793-X (220 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-18)
		A meromorphic function of several complex variables that is invariant under some discrete group of transformations
		(the complex projective line, or the Riemann sphere),
		Finally, one must mention the application of automorphic functions to the study of ordinary differential equations in a complex domain [12] and in the construction of solutions of algebraic equations of degrees higher than four.
	eom.springer.de /a/a014170.htm (1232 words)

	Amazon.com: Lectures on Vector Bundles over Riemann Surfaces. (MN-6): Books: Robert C. Gunning,R. C. Gunning (Site not responding. Last check: 2007-10-18)
		The author then generalizes the projective line results from chapter 2 to the case of a coherent analytic sheaf over an arbitary compact Riemann surface.
		After a lengthy discussion of how to extend a complex analytic line bundle to a complex analytic vector bundle of rank 2, the author in chapter 5 discusses how to determine which line bundles can be subbundles of a given vector bundle.
		Techniques from the theory of several complex variables are utilized without review to study the complex analytic equivalence of flat vector bundles, and he shows that every flat vector bundle is analytically equivalent to an 'irreducible' flat vector bundle.
	www.amazon.com /Lectures-Vector-Bundles-Riemann-Surfaces/dp/0691079986 (1314 words)

	riemann roch made easy
		For example, the complex projective line P^1 = C U {?}, the one point compactification of the complex numbers, is a compact Riemann surface of genus zero.
		Riemann's point of view was in the reverse order, since he considered as the basic object of study, a compact, connected branched cover of the projective line, and then proved such a manifold is a plane algebraic curve, possibly acquiring singular points from the plane mapping.
		A "divisor" is a finite formal linear combination of points, with integer coefficients D = nipi, on a smooth projective curve X. Then every meromorphic function f has an associated divisor div(f) consisting of the zeroes of f minus the poles of f, each counted with its appropriate multiplicity.
	www.physicsforums.com /showthread.php?t=85205 (3039 words)

	Orðasafn: L
		line at infinity sjóndeildarlína, = ideal line, = infinite line.
		line element lengdarfrymi, = element of length, = linear element.
		linear projective mapping varpmótun, línuleg varpmótun, = projective collineation, = projective mapping, = projective transformation, -> collineation, -> collineatory transformation, -> homography 2.
	www.hi.is /~mmh/ord/safn/safnL.html (2028 words)

	Complex Plane -- from Wolfram MathWorld
		The plane of complex numbers spanned by the vectors 1 and
		Every complex number corresponds to a unique point in the complex plane.
		The complex plane is sometimes called the Argand plane or Gauss plane, and a plot of complex numbers in the plane is sometimes called an Argand diagram.
	mathworld.wolfram.com /ComplexPlane.html (202 words)

	Doug's Expositions (Site not responding. Last check: 2007-10-18)
		In a more modern sense, projective geometry is important in algebraic geometry, complex analysis, and group theory.
		Acid test: if it is natural to add a line at infinity to a plane, then it is most natural to think of this as the real projective plane.
		Adding a point or line at infinity in geometry is one example of this, illustrating that there might be many ways to do this, with naturality depending on the context.
	www.math.columbia.edu /~zare/infinity.html (608 words)

	Polynomials, symmetry, and dynamics: An undertaking in aesthetics (Site not responding. Last check: 2007-10-18)
		This pair of lines corresponds to a pair of antipodal vertices of the dodecahedron.
		Each intersection is a real projective line as well as an "equatorial slice" of the associated complex projective line-a sphere.
		The green horizontal line corresponds to the intersection of the reflection plane R and the 36-line passing through the pair of green 72-points from the basin plot.
	members.tripod.com /vismath7/crass (5557 words)

	Real Projective Plane -- from Wolfram MathWorld
		(which, in the quotient space, is itself a projective line) corresponds to the line at infinity.
		Here, the projective plane is shown as a dashed circle, where lines continue on the opposite side of the circle.
		on the projective plane is the Petersen graph.
	mathworld.wolfram.com /RealProjectivePlane.html (310 words)

	[No title] (Site not responding. Last check: 2007-10-18)
		The relative index is shown to equal the spectral flow of a subelliptic complex which is shown, via deformation in the extended Heisenberg calculus to equal the index of the Dirac operator.
		We generalize the bending deformations of n-gon linkages in Euclidean space (points on the complex line) of Kapovich-Millson (JDG,vol.44,1996,pp.479-513) to points on complex projective m-space.
		This in turn indicates how to solve the Toda PDE by the factorization method in infinite dimensions, where the unitary group is replaced by the group of area-preserving transformations of the sphere.
	www.math.psu.edu /nistor/MEETINGS/talks.html (1116 words)

	Orðasafn: R
		real line 1 talnalína, talnaás, rauntalnalína, raunás, rauntalnaás, = continuum 2, = number axis, = number line, = numerical axis, = numerical line, = numeric line, = real axis.
		regression line aðhvarfslína, = line of regression, -> regression curve.
		Riemann sphere talnahvel Riemanns, = complex sphere, -> closed complex plane, -> complex projective line.
	www.hi.is /~mmh/ord/safn/safnR.html (2382 words)

	PlanetMath: complex projective line (Site not responding. Last check: 2007-10-18)
		is a projective variety called the complex projective line.
		Anyone with an account can edit this entry.
		This is version 4 of complex projective line, born on 2003-10-15, modified 2005-03-08.
	planetmath.org /encyclopedia/ComplexProjectiveLine.html (46 words)

	Karen's Webpage (Site not responding. Last check: 2007-10-18)
		The first four applets show different projections of inversions of chains in Heisenberg space, the boundary of CH ^ 2 (the complex hyperbolic plane) which has 3 real dimensions.
		The last applet shows inversions in CP1, the complex projective line, (a space with 2 real dimensions).
		This website and applets were created during a summer research project financed by an NSERC summer research grant.
	home.cc.umanitoba.ca /~umjohan1 (267 words)

	Polynomials, symmetry, and dynamics: An undertaking in aesthetics
		This pair of lines corresponds to a pair of antipodal vertices of the dodecahedron.
		The pictured "lines" are the images of small circles centered along the edges of the inner square.
		The green horizontal line corresponds to the intersection of the reflection plane R and the 36-line passing through the pair of green 72-points from the basin plot.
	www.mi.sanu.ac.yu /vismath/crass/index.html (5557 words)

	Quantum States
		Generally, the quantum state of a physical system is specified by a non-zero vector in a Hilbert space over the complex numbers (call it
		Mathematically, this is just the complex projective line.
		The correspondence between points on the complex projective line and points on the Riemann sphere is well-known; I will make use of it in a moment.
	math.ucr.edu /home/baez/lie/node10.html (289 words)

	Mathematics Colloquium: Chen Celebration, Spring 2005
		The Hopf Algebra of Iterated Integrals on the Complex Projective Line
		The iterated integrals on a manifold form a commutative algebra.
		We upgrade the iterated integrals on the complex projective line to more sophisticated motivic iterated integrals, and show that they form a (non-cocommutative) Hopf algebra.
	www.math.uiuc.edu /Colloquia/05SP/chen_celebration_apr14-05.html (116 words)

	Citebase - Derivations of an effective divisor on the complex projective line (Site not responding. Last check: 2007-10-18)
		Derivations of an effective divisor on the complex projective line
		In this paper we consider an effective divisor on the complex projective line and associate with it the module D consisting of all the derivations θ such that θ(I
		The module D is graded and free of rank 2; the degrees of its homogeneous basis, called the exponents, form an important invariant of the divisor.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0507323 (235 words)

	»»MN Reviews««
		The author then generalizes the projective line results from chapter 2 to the case of a coherent analytic sheaf over an arbitary compact Riemann surface.
		After a lengthy discussion of how to extend a complex analytic line bundle to a complex analytic vector bundle of rank 2, the author in chapter 5 discusses how to determine which line bundles can be subbundles of a given vector bundle.
		Techniques from the theory of several complex variables are utilized without review to study the complex analytic equivalence of flat vector bundles, and he shows that every flat vector bundle is analytically equivalent to an 'irreducible' flat vector bundle.
	www.financial-book-review.com /Low-grade/MN/MN_3.html (3459 words)

Try your search on: Qwika (all wikis)