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Longer titles found: Complex projective space (view), Real projective space (view), Quaternionic projective space (view), Gromov's inequality for complex projective space (view), Weighted projective space (view), Stunted projective space (view), Algebraic geometry of projective spaces (view), Fake projective space (view), Tropical projective space (view)

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alternate case: projective space

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projective plane and for any projective space defined arithmetically from a field or division ring; that includes any projective space of dimension greater than

A complete linear system | D | {\displaystyle |D|} is therefore a projective space. A linear system d {\displaystyle {\mathfrak {d}}} is then a projective

scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The

sphere like pointset (surface) in a projective space of dimension d ≥ 3. Simple examples in a real projective space are hyperspheres (quadrics). The essential

projective space over the dual numbers". mathoverflow.net. Retrieved 2017-04-16. "kt.k theory and homology - Grothendieck group for projective space over

Hermitian. Complex structures of Hn. Copies of complex projective space in quaternionic projective space. Dn (n ≥ 4) compact n(2n − 1) 0 Order 4 (cyclic when

dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced and classified by Zak (1993), who

University in 1948, the last with a dissertation titled A Class of Projective Space Curves written under Ingomar Hostetter. He then spent most of his career

Another useful class of examples come from ramified coverings of projective space. Given a homogeneous polynomial f ∈ C [ x 0 , … , x n ] {\displaystyle

his language the surfaces with a degree n embedding in n-dimensional projective space (del Pezzo 1887), which are the del Pezzo surfaces of degree at least

that has numerous applications is determining whether, and how, a projective space can be covered by pairwise disjoint subspaces which have the same dimension;

a reflectional symmetry: projective duality preserves symmetries a projective space, so dual curves have the same symmetry group. In this case both symmetries

and no three points are collinear (on a common line). An ovoid in a projective space is a set Ω of points such that: Any line intersects Ω in at most 2

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental

special case when the vector bundles are sums of line bundles over projective space. Thom (1957) pointed out that the fundamental class must be a polynomial

complete curve (for example, P1) and if f is a rational map from X to a projective space Pm, then f is a regular map X → Pm. In particular, when X is a smooth

dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor. Menger and Birkhoff

in four variables. The zeros form a cubic surface in 3-dimensional projective space. Salmon (1860) and Clebsch (1861, 1861b) studied the ring of invariants

tessellations of projective space in one less dimension. Defining k-dimensional projective polytopes in n-dimensional projective space is somewhat trickier

given by the intersection of the Grassmannian G(2, 4) (embedded in projective space P5 by Plücker coordinates) with a hypersurface. It is called a line

{\displaystyle \mathbb {Z} [x]/x^{n+1}.} In the case of infinite complex projective space, taking limits gives the answer Z [ x ] . {\displaystyle \mathbb {Z}

mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by Burkhardt (1890, 1891, 1892), with the maximum possible

threefold is a degree 4 hypersurface of dimension 3 in 4-dimensional projective space. Iskovskih & Manin (1971) showed that all non-singular quartic threefolds

projective geometry. A complex affine space can be obtained from a complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of

indecomposable vector bundles of rank n − 1 constructed on n-dimensional projective space Pn by Tango (1976) Tango, Hiroshi (1976), "An example of indecomposable

a Castelnuovo curve, studied by Castelnuovo (1889), is a curve in projective space Pn of maximal genus g among irreducible non-degenerate curves of given

Riemann surface, given by the image of the map from the Jacobian to projective space of dimension 2 g − 1 {\displaystyle 2^{g}-1} induced by theta functions

indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found by Horrocks (1978). Ancona, Vincenzo; Ottaviani, Giorgio (1995)

ring in three variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the

{\mathcal {O}}(1)} means that projective space is a Fano variety. Over the complex numbers, this means that projective space has a Kähler metric with positive

{\displaystyle z(s,t)={-3-(s^{2}+st+t^{2})(s+t) \over t(s^{2}+st+t^{2})-3}.} In projective space the Fermat cubic is given by w 3 + x 3 + y 3 + z 3 = 0. {\displaystyle

arrangement is the decomposition of the d-dimensional linear, affine, or projective space into connected cells of different dimensions, induced by a finite collection

class is effective, then it determines a rational map from V into projective space. This map is called the canonical map. The rational map determined

and in three-dimensional space it is a twisted cubic. Its closure in projective space is the rational normal curve. Moment curves have been used for several

algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities

Sperner's theorem, in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family.

The equivalence classes of such cubics form a three-dimensional real projective space and the subset of parabolic forms define a surface – the umbilic torus

2000s he wrote a series of papers on the divisible convex sets in projective space and periodic tilings by such sets. In 2011 he was awarded the Clay

exterior algebra to the bounded derived category of coherent sheaves on projective space. The importance of the notion rests on the suspicion that Koszul duality

space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Thurston

existence of a space model via an ovoid. An ovoid in a 3-dimensional projective space is a set of points, which a) is intersected by lines in 0, 1, or 2

complex projective space and there are many complex- analytic objects on it which are constructed from those on the complex projective space. When a complex-analytic

algebraic equations in some complex projective space. (Chow's Theorem says that any closed analytic subvariety of projective space is defined by algebraic equations

the projective cone is a conical surface; hence the name. Let X be a projective space over some field K, and R, S be disjoint subspaces of X. Let A be an

(V)} ), there is the natural identification (see Chern class#Complex projective space for example): Hom ⁡ ( l , V / l ) = T l P ( V ) {\displaystyle \operatorname

Robert; Van de Ven, Antonius (2012) [1984]. Topics in the geometry of projective space: Recent work of FL Zak (with an addendum by Fyodor Zak). DMV Seminar

1 plane in real projective space, and so translation in real Euclidean space can be represented as a shear in real projective space. Although a translation

Nonsingular surfaces of degree four in the real three-dimensional projective space. He has been a professor at the University of Strasbourg since 1991

In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if X is a projective scheme over a

Quadric#Normal_form_of_projective_quadricsusual definition of a conic in projective space uses a quadratic form. Another alternative definition of a conic uses

1991. His dissertation, On the Hyperplane Sections of a Variety in Projective Space, was supervised by Joe Harris. McKernan was the joint winner of the

Abelianisation Abelian variety, a complex torus that can be embedded into projective space Abelian surface, a two-dimensional abelian variety Abelian function

varieties in projective space. It gives a beautiful solution of an important problem. His theorem that a compact analytic variety in a projective space is algebraic

sheaf to the diagonal embedding of X over S. The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact

square root Reciprocity (projective geometry), a collineation from a projective space onto its dual space, taking points to hyperplanes (and vice versa)

that relates the bounded derived category of coherent sheaves on the projective space P n {\displaystyle \mathbb {P} ^{n}} and the stable category of graded

intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked

seen as follows: conics correspond to points in the five-dimensional projective space P 5 ; {\displaystyle \mathbf {P} ^{5};} requiring a conic to pass through

Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For further details see Schubert variety. The intersection

deformation limit of a quintic - i.e. a quintic hypersurface in the projective space C P 4 {\displaystyle \mathbb {CP} ^{4}} . The space C P 4 {\displaystyle

configuration, in geometry, a certain configuration in Euclidean space or projective space, consisting of two mutually inscribed tetrahedra Möbius–Kantor configuration

Grassmannian defined by conditions of incidence of a linear subspace in projective space with a given flag. For details see Schubert variety. According to Van

the complement of X(1) in 1-dimensional projective space. So it is the set of points (x:y) of projective space not fixed by the Frobenius map (x:y)→ (xq:yq)

homogeneous co-ordinates introduced initially to embed the space of lines in projective space P 3 {\displaystyle \mathbf {P} ^{3}} as a quadric in P 5 {\displaystyle

quadrangle dots, and the diagonal points he calls codots. The lines of the projective space are called straights, and in the quadrangle they are called connectors

of one sheet (non-degenerate quadric of index 2) in 3-dimensional projective space. Similar to the first two cases we get the (axiomatic) Minkowski plane

statement postulates that the group acts on the set of rays, that is, on projective space. Let (a, L) be an element of the Poincaré group (the inhomogeneous

distinct m-dimensional projective spaces contained in an n-dimensional projective space Rn. Let Pn−m−1 be an (n − m − 1)-dimensional subspace of Rn with no

moduli scheme of curves of genus g {\displaystyle g} embedded in some projective space. The Hilbert polynomial is given by P g ( n ) = ( 6 n − 1 ) ( g − 1

nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let D ( S ) {\displaystyle {\mathcal

, via the corresponding linear system of divisors, gives a map to projective space P ( H 0 ( V , K n ) ) = P P n − 1 {\displaystyle \mathbf {P} (H^{0}(V

separated). This can be used to compute the cohomology of line bundles on projective space. Leray's theorem Dimca 2004, 2.3.9. Godement 1973, Théorème 5.4.1.

to X is an isomorphism. An older notion is that a subvariety X of projective space is linearly normal if the linear system giving the embedding is complete

projective space. Therefore a polarised variety can be thought of as a projective variety together with a fixed embedding into some projective space C

sum of two copies of 3-dimensional projective space, and the product of S1 with two-dimensional projective space. The first two are mapping tori of the

is in the subspace r ⋅ (−no − ⁠1/2⁠n∞) = 0); then lift this into a projective space, by adjoining e– = 1, and identifying all points on the same ray from

Kirby–Siebenmann invariant: one is 2-dimensional complex projective space, and the other is a fake projective space, with the same homotopy type but not homeomorphic

projective line in the parameter space of curves, which is simply projective space. The Cayley–Bacharach theorem arises for high degree because the number

section j: W ⊂ X (i.e. W = X ∩ H, H some hyperplane in the ambient projective space), the maps j ∗ : H i ( X ) → H i ( W ) {\displaystyle j^{*}:H^{i}(X)\to

projective line in the parameter space of curves, which is simply projective space. The Cayley–Bacharach theorem arises for high degree because the number

system of points and plane sections of an ovoid in a three-dimensional projective space over GF(n). In 1962 he was an approved speaker (but not an invited

Non-affine examples come up when taking the stack quotient for weighted projective space/varieties. For example, the space P ( 2 , 3 ) {\displaystyle \mathbb

direction of periapsis and the current position of the body in the projective space. The projective anomaly is usually denoted by the θ {\displaystyle

endowed with the analytic topology coming from any embedding into a projective space P n ( k ) {\displaystyle \mathbb {P} ^{n}(k)} as a quasi-projective

a spread set, and this set of matrices can be used directly in the projective space P G ( 2 n − 1 , q ) {\displaystyle PG(2n-1,q)} to create a spread in

Suzuki groups as the symmetries of a certain ovoid in 3-dimensional projective space over a field of characteristic 2. Wilson (2010) constructed the Suzuki

continuous endomorphisms upon a Hilbert space H {\displaystyle H} , the projective space P ( H ) {\displaystyle \mathbf {P} (H)} of the Hilbert space H {\displaystyle

coordinate ring of the embedded abelian variety A, that is, set in a projective space according to a very ample L and its global sections. The graded commutative

can also be used with any field K and any dimension to construct the projective space Pn(K). Miyake, Modular forms, Springer, 2006, §1.1. This reference

group of order 2, then a model for BG is infinite-dimensional real projective space. It can be shown that if G is finite, then any CW-complex modelling

Geometry: Volume 2 Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties., Cambridge Mathematical

that every projective plane that can be embedded into a 3-dimensional projective space obeys Desargues' theorem. This three-dimensional realization of the

permutation groups. The projective general linear group acting on lines in a projective space of dimension at least 3 is a rank-3 permutation group. Several 3-transposition

permutation groups. The projective general linear group acting on lines in a projective space of dimension at least 3 is a rank-3 permutation group. Several 3-transposition

subset of its domain) in which w ≠ 0 {\displaystyle w\neq 0} ; in projective space this means we may take w = 1 {\displaystyle w=1} and therefore identify

line cannot be realized in the Euclidean plane, but forms a finite projective space known as the Fano plane. Because of this connection, the Kelly–Moser

compactification theorem, there is an embedding of this surface into a projective space. Hilbert modular form Picard modular surface Siegel modular variety

of Enriques surfaces. Take a surface of degree 6 in 3 dimensional projective space with double lines along the edges of a tetrahedron, such as w 2 x 2

degree −2 homogeneous extension of a function on the sphere and the projective space associated to Λ2R3 is identified with the space of all circles on the

Shafarevich, Igor (2013), Basic Algebraic Geometry 1: Varieties in Projective Space(3rd edition), Berlin, Springer-Verlag,ISBN 978-3-642-37955-0 Shafarevich

necessary. A topological sigma model targets infinite-dimensional projective space, and if such a thing could be defined it would have countably infinitely

0) is the 3-sphere, and the lens space L(2,1) is 3 dimensional real projective space. Lens spaces can be represented as Seifert fiber spaces in many ways

For example, if X = P S n {\displaystyle X=\mathbb {P} _{S}^{n}} is projective space, any coherent sheaf M {\displaystyle {\mathcal {M}}} on X {\displaystyle

{O}}(k)} over P n {\displaystyle \mathbb {P} ^{n}} is the weighted projective space minus a point O ( k ) = P ( 1 , … , 1 , k ) − { [ 0 : ⋯ : 0 : 1 ] }

space of possible quantum states of this particle, in this example the projective space P H {\displaystyle \mathrm {P} {\mathcal {H}}} associated with an 11-dimensional

American Mathematical Society in 2012. "Holomorphic maps into complex projective space omitting hyperplanes." Transactions of the American Mathematical Society

the relation between topological and holomorphic vector bundles on projective space. They solved the simplest unknown case, by showing that all rank 2

induced by the linear fractional transformation decomposes complex projective space into stable and unstable manifolds, with the horocycles appearing perpendicular

the set of vectors is a t {\displaystyle t} -design in the complex projective space C P n − 1 {\displaystyle \mathbb {CP} ^{n-1}} . Spherical design Quantum

motives R(j) in DM(k; R) for all integers j, such that the motive of projective space is a direct sum of Tate motives: M ( P k n ) ≅ ⊕ j = 0 n R ( j ) [

curves M g , n {\displaystyle M_{g,n}} , admits an embedding into a projective space, hence is a quasi-projective variety. This can be accomplished by looking

between homogeneous ideals of polynomials and algebraic subsets of a projective space, called the projective Nullstellensatz, that is analogous to the affine

H ∩ X, where X is a given smooth projective variety in the ambient projective space P N and H is a hyperplane. Then for i ≤ n = dim(X), the Lefschetz operator

Then V is normal at W, and the component of the transform of W is a projective space, which has dimension greater than W as predicted by Zariski's original

of such cubics under uniform scaling form a three-dimensional real projective space and the subset of parabolic forms define a surface – called the umbilic

Kevin; Hatey, Valérian; Tillich, Jean-Pierre (5 December 2023). "Projective Space Stern Decoding and Application to SDitH". arXiv:2312.02607 [cs.IT]

The Grothendieck Group of Algebraic Vector Bundles; Calculations of Affine and Projective Space Grothendieck Group of a Smooth Projective Complex Curve

of order 4, except for b=0 when it is a sum of two copies of real projective space, and |b|=1 when it is the lens space with fundamental group of order

amounts to describing the different ways a variety can be embedded into projective space. One answer is Kleiman's criterion (1966): for a projective scheme

defines an embedding of the outer space to some infinite dimensional projective space. In the second model an open simplex is given by all those R {\displaystyle

This may mean an intersection of n hypersurfaces in n-dimensional projective space. Archaic. linear Degree 1 lineo- Used to form compound adjectives such

V ) {\displaystyle \operatorname {\Gamma L} (V)} on the associated projective space P(V) yields the projective semilinear group, denoted P Γ L ⁡ ( V )