Projective space (${\mathbb{P}}^n$)

Projective space ${\mathbb{P}}^n$ is the set of one-dimensional subspaces of a vector space of dimension $n+1$, written in homogeneous coordinates $[x_0:\cdots :x_n]$.

An ordered projective frame consists of $n+2$ projective points such that any $n+1$ representative vectors are linearly independent. After mapping the first $n+1$ points to the coordinate axes, the last point has every coordinate nonzero; rescaling those axes sends it to $[1:\cdots :1]$. This proves existence and uniqueness of the projective transformation carrying one ordered frame to another. In the projective plane a frame consists of four points, no three collinear.

With the quotient convention, projective space represents the functor of surjections from a rank-$n+1$ free sheaf to a line bundle, modulo isomorphisms of the target compatible with the map. On an affine base the target may be a nontrivial rank-one projective module, so a tuple of elements generating the unit ideal is only the description when that target is trivial. Where the image of the $i$th basis vector generates the target, divide the other images by it to get the usual affine coordinates. Infinitesimal variations of the quotient form $\mathcal{H}om(\ker q,\mathcal{L})$ and give the Euler sequence.

A nonconstant morphism ${\mathbb{P}}^m\to {\mathbb{P}}^n$ with $m>n$ would have a positive-dimensional fiber of a morphism by the fiber dimension theorem. A target hyperplane avoiding its image point meets the positive-dimensional projective image, so its pullback is a nonempty divisor in ${\mathbb{P}}^m$ disjoint from that fibre. Every positive-dimensional projective subvariety meets every hypersurface: on its affine cone, a homogeneous equation cuts a positive-dimensional zero set containing a nonzero cone point. This contradiction proves constancy.

Hyperplanes in $\mathbb{P}(V)$ are kernels of nonzero linear functionals modulo scalar multiplication. Their parameter space is therefore $\mathbb{P}(V^{\ast })$, of the same dimension. In particular lines in a projective plane are parametrized by another projective plane.

For an irreducible projective variety $Y\subset \mathbb{P}(V)$, its projective dual is the closure of hyperplanes containing the embedded tangent space at some smooth point of $Y$. Over the complex numbers, biduality gives $(Y^{\ast }{)}^{\ast }=Y$. For a nondegenerate smooth curve in characteristic zero, the dual is an irreducible hypersurface; a general point corresponds to one ordinary tangency. Its defining incidence space consists of a point on the curve and a hyperplane containing its tangent line.

Over the complex numbers, an irreducible projective variety is recovered by applying projective dual variety twice. In its conormal incidence variety, pairs $(y,H)$ with $H$ tangent at $y$ become pairs $(H,y)$ under exchange of the two projective factors. At general smooth pairs, differentiating the incidence equation shows that the hyperplane defined by $y$ is tangent to the dual variety at $H$. Closure gives the same conormal variety with the factors exchanged, proving biduality. This is why the branch hypersurface of the Gauss map of a theta divisor determines a nonhyperelliptic canonical curve.

For a local ring $A$, every map from $\mathrm{Spec}A$ into projective space lands in one standard chart: the inverse image of a chart containing the closed point is the whole spectrum. This gives homogeneous coordinates with at least one unit entry, unique up to a common unit. For a nonlocal ring, coordinates may instead merely generate the unit ideal, and general maps can involve a nontrivial invertible sheaf quotient.

Quaternionic projective space is the space of right quaternionic lines in ${\mathbb{H}}^{n+1}$. It is $S^{4n+3}$ modulo the right action of unit quaternions, with one cell in each dimension $0,4,\dots ,4n$. It is a compact smooth manifold of real dimension $4n$; ${\mathbb{HP}}^1\cong S^4$. The quaternionic tautological line bundle and its sphere bundle give the multiplication in its cohomology ring of quaternionic projective space.

For the quaternionic tautological line bundle regarded as a complex rank-two bundle, take $q=c_2^{MU}$. Its ordinary image generates degree-four integral cohomology, and its square generates degree eight. The Atiyah-Hirzebruch spectral sequence collapses because the cells have dimensions $0,4,8$ and the coefficients are even. The classes $1,q,q^2$ lift a free basis of the associated graded module, and $q^3=0$ because the theory is connective and the space has dimension eight.

Coordinate inclusion of complex vectors sends a complex line to its quaternionic span. If two such vectors span the same quaternionic line, a nonzero complex coordinate forces their scalar ratio to be complex, so this map is injective. The pulled-back quaternionic tautological line bundle splits as $L\oplus \overline{L}$ with complex first Chern classes $-x,x$. Its top Chern class, and thus Euler class, is $-x^2$. With $u$ the negative Euler generator, $i^{\ast }u=x^2$ and $i^{\ast }(u^k)=x^{2k}$. The map on infinite projective spaces is injective on cohomology; a finite inclusion loses the degrees with $2k>n$.

The cell structure gives one integral cohomology generator in each degree divisible by four through $4n$. The Gysin sequence of a sphere bundle for the quaternionic tautological line bundle makes multiplication by its Euler class an isomorphism between successive such degrees. Choose $u$ to be the negative of that Euler class; its powers generate the displayed cohomology ring.

Points in ${\mathbb{P}}^n$ are in general linear position when no $k+1$ of them lie in a projective subspace of dimension less than $k$ for $k\le n$. Equivalently, every selection of at most $n+1$ representing vectors is linearly independent.

Distinct lines in the projective plane are in general position when no three are concurrent. Their union is a reduced plane curve with exactly $\left(\genfrac{}{}{0ex}{}{d}{2}\right)$ singular points, all ordinary pairwise crossings, if there are $d$ lines. An explicit family is $y=t_{i}x+t_i^{2}z$ for distinct complex $t_i$: the intersection of the $i$th and $j$th lines is $[-t_i-t_j:-t_i{t}_j:1]$, and substituting it into a third equation gives $-(t_k-t_i)(t_k-t_j)\ne 0$.

The degree-$d$ Veronese map sends a projective point to the list of all degree-$d$ monomials in its homogeneous coordinates. It embeds ${\mathbb{P}}^n$ into a higher-dimensional projective space.

For a field $k$,This nonvanishing obstructs affine covers of ${\mathbb{P}}_k^n$ having only $n$ members.

For every integer $d$, the cohomology of a twisting sheaf on projective space isThus all intermediate cohomology vanishes. The top line follows from Serre duality and ${\omega }_{{\mathbb{P}}^n}\cong \mathcal{O}(-n-1)$.

For $m\ge 0$, these sheaf cohomology dimensions follow from the vanishing for $\mathcal{O}$ and the hyperplane exact sequence for twisting sheaves. Induct on dimension and then on $m$ in $0\to \mathcal{O}(m-1)\to \mathcal{O}(m)\to i_{\ast }{\mathcal{O}}_{{\mathbf{P}}^{n-1}}(m)\to 0$. The long exact sequence in sheaf cohomology gives higher-degree vanishing and the recurrence $h_n^0(m)-h_n^0(m-1)=\left(\genfrac{}{}{0ex}{}{n-1+m}{n-1}\right)$. Starting with $h_n^0(0)=1$ yields the formula; the corresponding homogeneous monomials form a basis of global sections.

On the projective line, an infinite direct sum of copies of $\mathcal{O}(-1)$ is a quasi-coherent sheaf but not a coherent sheaf: at any point its residue-vector-space dimension is infinite. Its global sections vanish because each summand has no sections, and sections commute with direct sums for the finite two-chart equalizer. Its direct image sheaf on $\mathrm{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.

The only regular functions defined on all of a nonempty projective space over a field $k$ are the constants:For ${\mathbb{P}}^1$, this also follows by gluing $k[t]$ and $k[t^{-1}]$ inside $k[t,t^{-1}]$: their intersection is $k$.

A projective point is an equivalence class $[x_0:\cdots :x_n]$ of nonzero coordinate vectors under multiplication by a nonzero scalar.

The homogeneous coordinates $[x_0:\cdots :x_n]$ of a projective point are defined only up to simultaneous multiplication by a nonzero scalar.

A coordinate point of ${\mathbb{P}}^n$ is represented by a standard basis vector, so exactly one of its homogeneous coordinates is nonzero.

The projective plane is the two-dimensional projective space ${\mathbb{P}}^2$.

For two triples of distinct points on two projective lines, the intersections $r_{ij}=p_i{q}_j\cap p_j{q}_i$ are collinear when the configuration is defined. Normalize a projective frame so $p_1=[0,1,0]$, $p_2=[1,1,0]$, $q_1=[0,0,1]$, $q_2=[1,0,1]$, and write $p_3=[1,\alpha ,0]$, $q_3=[1,0,\beta ]$. Then representatives of the three intersections are $(1,1,1)$, $(1,\alpha ,\beta )$, and $(\alpha \beta -1,\alpha (\beta -1),\beta (\alpha -1))$. The last is $\alpha \beta$ times the first minus the second, proving collinearity by linear dependence. This gives a coordinate proof over any field when the normalizations are available.

In the real projective plane, represent a projective point by a nonzero vector and a projective line by its nonzero coefficient covector, identified with a vector in chosen coordinates. The join of distinct projective points represented by $p,q$ has coefficient vector $p\times q$, using the cross product. It is nonzero because the representatives are linearly independent, and its dot product with each representative vanishes. Dually, two distinct projective lines with coefficients $\ell ,m$ meet in the projective point represented by $\ell \times m$. Scaling an input scales the output and leaves the projective object unchanged. This makes joins, intersections, concurrency, and collinearity exact calculations with homogeneous coordinates.

For two projective triangles in a projective plane over a field, perspectivity from a point implies perspectivity from a line. The usual nondegenerate form assumes that the corresponding-side intersections are well defined and distinct. Its dual is the converse, by point-line duality. The result holds in the real projective plane; it is not an axiom valid in every abstract incidence plane.

One linear algebra proof explains the collinearity. Choose a representative $o$ of the center. Rescale the paired representatives to $p^{\prime }=p+\alpha o$, $q^{\prime }=q+\beta o$, and $r^{\prime }=r+\gamma o$, with nonzero coefficients in a nondegenerate configuration whose vertices differ from the center. The first corresponding-side intersection is represented by $x=\beta p-\alpha q=\beta p^{\prime }-\alpha q^{\prime }$. Similarly choose $y=\gamma q-\beta r$ and $z=\alpha r-\gamma p$. Then $\gamma x+\alpha y+\beta z=0$, so the three representatives are linearly dependent. Their projective points are consequently collinear.

Two projective triangles are in perspectivity from a line when the three intersections of corresponding sides lie on one projective line, called their axis of perspectivity. In the nondegenerate setting this is dual to perspectivity from a point under point-line duality.

Two projective triangles are in perspectivity from a point when the three projective lines joining corresponding vertices meet in one projective point, called their center of perspectivity.

A projective triangle is an ordered triple of noncollinear projective points in a projective plane, together with their three joining projective lines. In the real projective plane this means that three chosen representative vectors are linearly independent. Pairing the vertices of two such triangles determines corresponding sides and the three corresponding-vertex joins.

After choosing the affine chart $z\ne 0$ in the projective plane, its complement $z=0$ is the projective line at infinity. Affine parallel lines meet at the same projective point on this line, corresponding to their common direction. The choice of this line is part of the affine chart, not an intrinsic distinction preserved by every projective transformation.

A complete quadrangle consists of four projective points in the projective plane, no three collinear, and all six joining projective lines. Its three diagonal points are the intersections of opposite joining lines. Over a field of characteristic different from two, the line joining two diagonal points cuts either side through the third diagonal point in a pair of harmonic conjugate points. In a projective frame this follows by using the coordinate points and $[1:1:1]$ and computing the intersections.

A projective conic is the zero locus in a projective plane of a nonzero homogeneous degree-two polynomial. The equation is considered up to multiplication by a nonzero scalar. Over the complex numbers, a symmetric matrix for the quadratic form has rank three, two, or one: these give a nonsingular conic, two distinct projective lines, or one line counted twice. Diagonalization of a quadratic form and rescaling its nonzero diagonal entries prove this classification. The smooth model $XZ-Y^2=0$ is parametrized by the degree-two Veronese map $[s:t]\mapsto [s^2:st:t^2]$.

Four projective points in general linear position in the complex projective plane impose four independent linear conditions on the six coefficients of a quadratic polynomial. To see independence, put them at the three coordinate points and $[1:1:1]$ using a projective frame. The remaining equations are $Axy+Bxz+Cyz=0$ with $A+B+C=0$, a two-dimensional vector space. Projectivizing its nonzero elements gives a pencil. It has nonsingular bases, for example the forms with coefficient triples $(1,1,-2)$ and $(1,-2,1)$, because the determinant of the symmetric coefficient matrix is $ABC/4$. Its three singular members are exactly the three pairs of lines obtained by partitioning the four base points into two pairs.

A point $[a:b:c]$ of the projective plane corresponds to the dual line $aX+bY+cZ=0$. A line given by those coefficients corresponds back to the point $[a:b:c]$. Incidence is reversed: $p\in \ell$ if and only if ${\ell }^{\ast }\in p^{\ast }$. Thus collinear point triples become concurrent line triples, preserving the essential combinatorics of incidence geometry.

Dualize a noncollinear finite point set in the real projective plane to projective lines. Their intersection vertices and consecutive line segments form an embedded graph. A vertex incident to $k$ lines has graph degree $2k$ and corresponds to a primal line containing $k$ points. Ordinary lines correspond to degree-four vertices. The faces are polygons with at least three sides.

A good dual-arrangement edge has degree-six endpoints and two triangular adjacent faces. Other edges are bad. The Euler defect identity for a projective line arrangement bounds the number of bad edges by $16t_2$: charge them to degree-four or higher-degree vertices and to nontriangular faces. This turns a count of ordinary lines into quantitative control of local triangular geometry.

A safe dual-arrangement edge is one for which every edge on a path of length at most two from either endpoint is a good dual-arrangement edge. This stronger local condition supplies a two-cell-thick triangular strip. Only a bounded multiple of the original bad edges can be unsafe: trace a shortest path to the first bad edge and reverse it through degree-six intermediate vertices.

In an embedded arrangement graph, an edge whose fixed-radius neighbourhood meets a bad edge can be charged to the first such edge along a shortest path. All intervening edges are good dual-arrangement edges, so their endpoints have bounded degree. The number of possible reversed paths is bounded in terms of the fixed radius alone. Therefore thickening a defect set by a fixed radius preserves an $O(t_2)$ edge count.

Let $t_k$ count vertices where $k$ lines meet and $f_j$ count faces with $j$ sides in a nonpencil projective line arrangement. Using Euler characteristic one, $v=\sum t_k$, $e=\sum k{t}_k$, and $2e=\sum j{f}_j$ gives $t_2=3+{\sum }_{k\ge 4}(k-3)t_k+{\sum }_{j\ge 4}(j-3)f_j$. Thus few double vertices control both high-multiplicity vertices and nontriangular faces. Discarding the face term gives the usual arrangement form of the ordinary-line inequality.

The Fano plane is the projective plane over the finite field ${\mathbb{F}}_2$. It has seven points and seven lines, each line contains three points, and two distinct lines meet in one point. A line-point incidence matrix $N$ therefore satisfies $N{N}^{\mathsf{T}}=2I+J$, making it invertible over the real numbers and useful for the transplantation theorem.

Adjoin the inverse-transpose automorphism to the collineation group of the Fano plane. Its maximal subgroups other than the collineation group are unordered incident-pair stabilizers, unordered nonincident-pair stabilizers and Sylow 7-normalizers, of orders 16, 12 and 42.

There are 21 incident point-line pairs and 28 nonincident pairs in the Fano plane, with $G{L}_3(2)$ transitive on each type. Orbit-stabilizer theorem gives the two orders as $168/21=8$ and $168/28=6$.

An invertible linear map of the underlying vector space induces a projective linear transformation of ${\mathbb{P}}^n$; scalar multiples induce the same transformation.

A projective variety is a closed algebraic subvariety of some projective space.

The complete system of quadratic forms maps ${\mathbb{P}}^2$ to a smooth surface of degree four in ${\mathbb{P}}^5$. The six coordinates are $x_0^2,x_0{x}_1,x_0{x}_2,x_1^2,x_1{x}_2,x_2^2$. It occurs as a degree-two image of a K3 surface and, after taking a projective cone, in the plane-quintic exception to the K3 linear system dichotomies.

A projective embedding realizes a variety as a closed subvariety of projective space. A very ample line bundle gives such an embedding through its global sections.

Suppose a nonconstant morphism of varieties $f:X\to Z$ has fiber $Y$ over $z_0$, where $X$ is normal, connected and projective. Choose an affine neighborhood $U$ of $z_0$ and a regular function on $U$ nonconstant on $f(X)\cap U$. Its pullback is a nonconstant rational function on $X$, so its pole Weil divisor is nonzero. A pole-free rational function on a normal variety is regular, and global regular functions on a connected projective variety are constant. The pole divisor avoids $f^{-1}(U)$, hence avoids $Y$. No projectivity assumption on $Z$ is needed.

A projective completion of an affine variety $X$ is a projective variety containing $X$ as a dense open subvariety. Homogenizing affine defining equations and taking their projective zero set gives the projective closure, although it may add singularities at infinity.

The homogenization of a polynomial $f(x_1,\dots ,x_n)$ of total degree $d$ is the homogeneous polynomialIts projective zero set is the projective closure of the affine hypersurface defined by $f$.

If projective varieties $X,Y\subseteq {\mathbb{P}}^n$ have dimensions $r,s$ and $r+s\ge n$, then $X\cap Y$ is nonempty and every component has dimension at least $r+s-n$.

A projective curve is a projective algebraic curve. A projective curve is a one-dimensional projective variety.

A rational normal curve in ${\mathbb{P}}^n$ is the image of the degree-$n$ Veronese map from ${\mathbb{P}}^1$. After projective changes of coordinates it has parametrization

A smooth projective curve is a nonsingular projective curve. A smooth projective curve is a nonsingular projective variety of dimension one.

A nonconstant rational function on a smooth projective curve extends to a finite morphism to the projective line. The inverse images of its two standard affine charts are affine varieties covering the curve. The resulting Čech cochain complex has only degrees zero and one, so every coherent sheaf has zero sheaf cohomology in degrees at least two.

For a smooth projective curve with $H^0(C,{\mathcal{O}}_C)=k$, its genus is ${\dim }_k{H}^1(C,{\mathcal{O}}_C)$, equivalently ${\dim }_k{H}^0(C,{\Omega }_C^1)$ by Serre duality. It equals the arithmetic genus $1-\chi ({\mathcal{O}}_C)$. Without the condition on constants, the general Riemann-Roch theorem instead uses $\chi ({\mathcal{O}}_C)=h^0({\mathcal{O}}_C)-h^1({\mathcal{O}}_C)$.

Over an algebraically closed field, choose a point $P$ on the smooth projective curve. The Riemann-Roch theorem gives $\dim L(P)=2$: the complementary canonical divisor term has negative degree. A nonconstant function in $L(P)$ has exactly one simple pole, so it defines a degree-one morphism of algebraic curves to the projective line. A degree-one map between smooth projective curves is an isomorphism. The existence of a rational point is essential over a field which is not algebraically closed.

Over the algebraic closure of ${\mathbb{F}}_3$, the displayed cubic is smooth. Its three partial derivatives are $Z^2$, $2YZ$, and $Y^2+2XZ$; their simultaneous vanishing would give $Z=Y=0$, and the equation would then give $X=0$, impossible in projective space. Its affine equation $y^2=x^3-x$ is irreducible because the odd-degree polynomial on the right is not a square in the rational function field.

Over an algebraically closed field, a smooth projective rational curve is isomorphic to ${\mathbb{P}}^1$. Its line bundles are determined by degree, its degree-zero line bundle is trivial, and $H^1({\mathbb{P}}^1,\mathcal{O})=0$. An integral projective curve of arithmetic genus zero is automatically smooth and rational by the normalization genus formula.

At every point of a smooth algebraic curve, the local ring is a discrete valuation ring. Its valuation measures the order of vanishing of a rational function at that point.

At a point of a smooth algebraic curve, its local ring is a discrete valuation ring. A local parameter is a generator $t$ of its maximal ideal, equivalently an element of valuation one. It measures the order of a zero of a function or pole by expressing functions as powers of $t$ times units. Over the complex numbers its analytic counterpart is a local coordinate vanishing simply at the point. For an elliptic curve in a Weierstrass equation of an elliptic curve, $t=-x/y$ is the standard parameter at the identity.

Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.

Let $\nu :\tilde{C}\to C$ be the normalization of a nodal curve. Its rational inverse $C⇢\tilde{C}$ cannot extend over the node: the two branches give two distinct points of $\tilde{C}$, whereas a morphism can assign only one image to the node.

For $X=V_{+}(I)\subseteq {\mathbb{P}}^n$, the homogeneous coordinate ring is the graded ring $k[x_0,\dots ,x_n]/I$.

The product of projective varieties is projective via the Segre embedding. Dimensions add under products, and the product of smooth varieties is smooth.

The coordinate projections from a product send $(x,y)$ to $x$ and $y$, respectively. For products of projective varieties, they are morphisms.

The Segre embedding sendsand realizes ${\mathbb{P}}^m\times {\mathbb{P}}^n$ as a projective variety.

For the Segre embedding of ${\mathbb{P}}^n\times {\mathbb{P}}^m$, the homogeneous coordinate ring is the algebra generated by $x_i{y}_j$, assigning degree one to every generator. Its degree-$t$ piece consists of all polynomials bihomogeneous of bidegree $(t,t)$: pair the $t$ factors of an $x$-monomial with the $t$ factors of a $y$-monomial to express any such monomial as a product of generators. Counting the independent monomials gives the displayed Hilbert polynomial, agreeing with the Hilbert function for every integer $t\ge 0$. Its polynomial degree is $n+m$, giving the variety's dimension. Its leading coefficient is $1/(n!m!)$, giving embedded projective degree $(n+m)!/(n!m!)=\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$. The two uses of degree therefore describe different quantities.

A smooth quadric surface is a smooth two-dimensional quadric hypersurface. Over an algebraically closed field of characteristic other than two, every smooth quadric surface in ${\mathbb{P}}^3$ becomesafter a projective linear transformation. The Segre embedding identifies it with ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

For $X=\{x_0{x}_1=x_2{x}_3\}$ and $P=[1:0:0:1]$, the displayed map is defined on $X\setminus \{P\}$. Its fibre over $[u:v:w]$ has equation $(u-v)\lambda +uw=0$ on an affine line. Away from $u=v$ the fibre is one reduced point. On that line it is empty except at $[0:0:1]$ and $[1:1:0]$, where it is a punctured ruling line, hence an affine line. The two exceptional lines are the rulings of a smooth quadric surface through $P$.

Over an algebraically closed field, remove a smooth hyperplane conic $C$ from a smooth quadric surface $Q$. The rulings of a smooth quadric surface generate $\mathrm{Pic}(Q)={\mathbb{Z}}^2$, and $C$ has class $(1,1)$. Picard-group localization on a smooth variety gives the quotient by $\mathbb{Z}(1,1)$. The restricted ruling bundle ${\mathcal{O}}_Q(1,0)$ is a generator, exhibiting a nontrivial line bundle on an affine surface.

Over an algebraically closed field, a smooth four-variable projective quadratic equation can be written $x_0{x}_3-x_1{x}_2=0$, including in characteristic two. Its quadratic form splits into two hyperbolic planes: choose an isotropic vector and its paired vector, adjust the latter to be isotropic, and repeat on the orthogonal complement. In characteristic two, smoothness forces the alternating polar form in dimension four to have full rank, since a positive-dimensional radical has a nonzero isotropic vector over the algebraically closed field and would give a singular point. The Segre embedding $([s_0:s_1],[t_0:t_1])\mapsto [s_0{t}_0:s_0{t}_1:s_1{t}_0:s_1{t}_1]$ then gives the isomorphism. Each matrix of coordinates has rank one, which supplies the inverse pair of projective factors.

Under $Q\cong {\mathbb{P}}^1\times {\mathbb{P}}^1$, the fibres of the two coordinate projections are the two rulings of the smooth quadric surface. Two distinct fibres in one ruling are disjoint projective lines, while one fibre from each ruling meets in one point.

For distinct $p,q\in {\mathbb{P}}^1$, the curves ${\mathbb{P}}^1\times \{p\}$ and ${\mathbb{P}}^1\times \{q\}$ are disjoint, smooth, and projective.