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 $[x_0:\cdots :x_n]$.

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 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.

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.

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 ${\omega }_{{\mathbb{P}}^n}\cong \mathcal{O}(-n-1)$.

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 ${\mathbb{P}}^2$.

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.

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.

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 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

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.

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.

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

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.