04.07.01 · algebraic-geometry / projective

Projective space

shipped3 tiersLean: partial

Anchor (Master): Hartshorne §II; Vakil; Grothendieck-Dieudonné EGA II; Eisenbud-Harris

Intuition [Beginner]

Projective space $P^{n}$ is the space of lines through the origin in $(n + 1)$ -dimensional space. Equivalently: ordinary $n$ -dimensional space, plus a sphere of "points at infinity" — one point at infinity for every direction you might travel.

Why care? Many natural geometric statements that have annoying exceptions in ordinary affine space become clean and exception-free in projective space. Two distinct lines in the plane meet at exactly one point — except when they're parallel; in projective space, "parallel" means "meeting at the same point at infinity," and the statement holds with no exception. The number of intersections of curves of given degrees is constant in projective space (Bézout's theorem), again with no exception.

Projective space is the natural home for algebraic geometry: smooth projective varieties are compact (in the analytic topology over $C$ ), so they admit clean cohomological invariants, finite-dimensional function spaces, and the full power of Riemann-Roch. Affine space is a punctured version of projective space; projective space is the cleaner and more symmetric object.

Visual [Beginner]

A line through the origin in $R^{3}$ corresponds to a single point of $P^{2} (R)$ ; the lines are the points of projective space.

$A bundle of lines through the origin in three-space, each representing a single point of the projective plane $\mathbb{P}^2$.$

Worked example [Beginner]

The projective line $P^{1}$ over the real numbers: points are equivalence classes $[x : y]$ of pairs $(x, y) \neq = (0, 0)$ , where $[x : y] \sim [λ x : λ y]$ for any $λ \neq = 0$ .

Two charts cover $P^{1}$ :

$U_{0} = {[x : y] : x \neq = 0}$ with coordinate $u = y / x$ . This sends $[x : y]$ to $u$ and recovers the real line $R$ as ${[1 : u] : u \in R}$ .
$U_{1} = {[x : y] : y \neq = 0}$ with coordinate $v = x / y$ .

The transition function on the overlap $U_{0} \cap U_{1}$ is $v = 1/ u$ . Topologically, $P^{1} (R)$ is a circle (real line plus one point at infinity), and $P^{1} (C)$ is the Riemann sphere.

The projective plane $P^{2}$ over $R$ is a 2-dimensional surface obtained from the sphere $S^{2}$ by identifying antipodal points. Over $C$ , $P^{2} (C)$ is a smooth complex surface (real dimension 4) — a basic and important compact complex manifold.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $k$ be a field. The projective space $P_{k}^{n}$ has three equivalent constructions.

(1) Set-theoretic / quotient construction. As a set,

P^{n} (k) := (k^{n + 1} ∖ {0}) / k^{\times},

where $k^{\times}$ acts by scalar multiplication. A point is denoted $[x_{0} : x_{1} : \dots : x_{n}]$ , the equivalence class of $(x_{0}, \dots, x_{n})$ .

(2) Open-affine cover. Cover $P_{k}^{n}$ by $n + 1$ open subsets $U_{i} := {[x_{0} : \dots : x_{n}] : x_{i} \neq = 0}$ for $i = 0, \dots, n$ . Each $U_{i}$ is isomorphic to affine $n$ -space:

U_{i} ≅ A_{k}^{n}, [x_{0} : \dots : x_{n}] \mapsto (x_{0} / x_{i}, \dots, x_{i} / x_{i}, \dots, x_{n} / x_{i}) .

The transition functions on overlaps $U_{i} \cap U_{j}$ are rational maps of the form $u_{l} \mapsto u_{l} / u_{j}$ — algebraic regular functions on the open subsets.

(3) Proj construction (scheme-theoretic). As a scheme,

P_{k}^{n} := Proj k [x_{0}, x_{1}, \dots, x_{n}],

where $Proj S$ for a graded ring $S$ is constructed analogously to $Spec R$ but with homogeneous prime ideals (those not containing the irrelevant ideal $S_{+} = ⨁_{d > 0} S_{d}$ ). Topologically, points of $P_{k}^{n}$ are homogeneous prime ideals of $k [x_{0}, \dots, x_{n}]$ other than the irrelevant ideal.

(4) Functor of points. Most modern: $P_{k}^{n}$ represents the contravariant functor

P_{k}^{n} (R) := {(L, φ) : L invertible R -module, φ : R^{n + 1} ↠ L} / \sim,

i.e., surjections from $R^{n + 1}$ onto invertible $R$ -modules, modulo isomorphism. This characterises $P_{k}^{n}$ as the moduli space of line bundles together with $n + 1$ generating sections.

Twisting sheaves. Define the twisting sheaf $O (d)$ on $P_{k}^{n}$ for each $d \in Z$ as the sheaf associated to the graded module $k [x_{0}, \dots, x_{n}] (d)$ (degree shift by $d$ ). For $d \geq 0$ :

H^{0} (P_{k}^{n}; O (d)) = k [x_{0}, \dots, x_{n}]_{d} (homogeneous polynomials of degree d) .

For $d < 0$ , $H^{0} = 0$ .

Properties of $P_{k}^{n}$ :

Connected, irreducible, reduced — $P_{k}^{n}$ is a smooth projective variety.
Dimension: $dim P_{k}^{n} = n$ .
Picard group: $Pic (P_{k}^{n}) = Z$ , generated by $O (1)$ .
Cohomology of $O (d)$ : foundational computation, given by the Serre vanishing-and-formula:

H^{i} (P_{k}^{n}; O (d)) = ⎩ ⎨ ⎧ k [x_{0}, \dots, x_{n}]_{d} k [x_{0}, \dots, x_{n}]_{- d - n - 1}^{\lor} 0 i = 0, d \geq 0 i = n, d \leq - n - 1 otherwise.

Canonical sheaf: $ω_{P_{k}^{n}} = O (- n - 1)$ .
Automorphism group: $Aut (P_{k}^{n}) = PGL_{n + 1} (k)$ .

Key theorem with proof [Intermediate+]

Theorem (Picard group of projective space). For an algebraically closed field $k$ and $n \geq 1$ , $Pic (P_{k}^{n}) = Z$ , generated by $O (1)$ . Every line bundle on $P_{k}^{n}$ is isomorphic to $O (d)$ for a unique $d \in Z$ .

Proof sketch. Use the cover $U_{i} = {x_{i} \neq = 0}$ , on each of which $P^{n}$ restricts to $A^{n}$ . The Picard group of $A_{k}^{n}$ vanishes (every line bundle on affine space vanishes — by Quillen-Suslin / the polynomial ring case of $Pic (Spec R) = 0$ when $R$ is a polynomial ring over a field). So a line bundle $L$ on $P^{n}$ vanishes on each $U_{i}$ .

A line bundle $L$ on $P^{n}$ is then determined by its transition functions on overlaps $U_{i} \cap U_{j}$ — invertible regular functions $g_{ij} \in O^{\times} (U_{i} \cap U_{j})$ . Compute $O^{\times} (U_{i} \cap U_{j}) = k^{\times} \cdot (x_{j} / x_{i})^{Z}$ — Laurent monomials in $x_{j} / x_{i}$ with leading $k^{\times}$ scalar.

The cocycle condition $g_{ij} g_{j k} = g_{ik}$ on triple overlaps forces $g_{ij} (x) = c_{ij} \cdot (x_{j} / x_{i})^{d}$ for a fixed integer $d$ and constants $c_{ij} \in k^{\times}$ satisfying $c_{ij} c_{j k} = c_{ik}$ — but these constants are coboundaries (any cocycle of $k^{\times}$ -valued data on a contractible cover is a coboundary), so they can be normalised to 1. The remaining data is the integer $d$ , the degree of $L$ .

So $L ≅ O (d)$ for unique $d \in Z$ . $□$

This computation is the foundational first nonzero Picard-group calculation in algebraic geometry. Every later Picard-group computation builds on similar Čech-cocycle reasoning.

Bridge. The construction here builds toward 04.05.05 (ample and very ample line bundle), where the same data is upgraded, and the symmetry side is taken up in 04.07.02 (blowup). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 5 (medium, conceptual).

Explain the Plücker embedding of the Grassmannian $Gr (2, 4)$ into $P^{5}$ .

Hint

Send a 2-plane $W \subset k^{4}$ to the wedge $w_{1} \land w_{2} \in Λ^{2} k^{4} ≅ k^{6}$ .

Answer

$Gr (2, 4)$ is the variety of 2-dimensional subspaces of $k^{4}$ . Choose a basis $w_{1}, w_{2}$ of $W$ . The exterior product $w_{1} \land w_{2} \in Λ^{2} k^{4}$ is well-defined up to scalar (changing basis multiplies by a scalar = $det$ of change-of-basis matrix). So $W \mapsto [w_{1} \land w_{2}]$ defines a map $Gr (2, 4) \to P (Λ^{2} k^{4}) = P^{5}$ .

The image is cut out by the Plücker relations: a vector $\sum p_{ij} e_{i} \land e_{j} \in Λ^{2} k^{4}$ is decomposable as $w_{1} \land w_{2}$ iff $p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0$ . So $Gr (2, 4)$ is a smooth quadric hypersurface in $P^{5}$ .

This Plücker embedding generalises: every Grassmannian $Gr (k, n)$ embeds in $P (Λ^{k} k^{n})$ as a smooth subvariety, providing a uniform way to study these spaces.

Exercise 6 (hard, proof).

State Bézout's theorem for plane curves and explain why projective space is the natural setting.

Hint

Affine intersections can have lower count due to "intersections at infinity" or parallel curves.

Answer

Bézout's theorem (plane curves). Let $C, D \subset P_{k}^{2}$ be plane projective curves of degrees $d, e$ respectively, with no common irreducible component. Then their intersection $C \cap D$ has exactly $d e$ points, counted with multiplicity.

In affine $A^{2}$ , this can fail: two lines (degrees 1 and 1) "should" meet in $1 \cdot 1 = 1$ point, but parallel lines meet in zero affine points. Adding the line at infinity to make $P^{2}$ , parallel lines meet at the point at infinity in their common direction — restoring the count.

More generally: a smooth conic (degree 2) and a line (degree 1) "should" meet in 2 points; in $A^{2}$ they can have 0, 1, or 2 intersection points depending on geometry. In $P^{2}$ , the count is always exactly 2 (with multiplicity), thanks to:

Compactness: $P^{2}$ is compact, so all "limit" intersections are captured.
Multiplicity: tangencies and singular intersections are weighted properly.
No exceptions for "infinity": every direction has a single point at infinity.

Bézout's theorem generalises to intersection numbers on smooth projective varieties, the foundation of intersection theory (Fulton, Eisenbud-Harris).

Exercise 7 (hard, conceptual).

Explain how $P^{n}$ represents the functor of "line bundles with $n + 1$ generating sections."

Hint

A point of $P^{n} (R)$ is the data of a surjection $R^{n + 1} \to L$ where $L$ is invertible.

Answer

For a ring $R$ , a morphism $φ : Spec R \to P_{k}^{n}$ is equivalently:

(scheme map) A morphism of schemes $Spec R \to P_{k}^{n}$ over $Spec k$ .
(functorial point) An equivalence class $(L, s_{0}, \dots, s_{n})$ where $L$ is an invertible $R$ -module, $s_{0}, \dots, s_{n} \in L$ are global sections that generate $L$ (i.e., the map $R^{n + 1} \to L$ , $(a_{0}, \dots, a_{n}) \mapsto \sum a_{i} s_{i}$ , is surjective), modulo isomorphism of the data.

Concretely: pulling back $O (1)$ gives the line bundle $L = φ^{*} O (1)$ , and the global sections $x_{0}, \dots, x_{n} \in H^{0} (P^{n}; O (1))$ pull back to generators $s_{i} = φ^{*} x_{i}$ of $L$ . The surjectivity (no common zeroes) corresponds to the morphism being well-defined everywhere.

This functorial description is the modern way to think about $P^{n}$ : it represents the moduli problem "line bundle equipped with $n + 1$ generators." Higher Grassmannians and flag varieties have analogous moduli interpretations, and this perspective generalises to all projective bundles and varieties.

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has Projectivization (the set-theoretic projective space construction) and AlgebraicGeometry.ProjectiveSpace (scheme-theoretic via Proj), with much of the basic API.

[object Promise]

Advanced results [Master]

Cohomology of line bundles on $P^{n}$ . The complete computation of $H^{i} (P_{k}^{n}; O (d))$ for all $i, d$ — given in the formal-definition section — is the foundational cohomology calculation in algebraic geometry. Every projective-variety cohomology computation eventually reduces to this via long exact sequences.

Hilbert polynomial. Every coherent sheaf $F$ on a closed subscheme $X \subset P^{n}$ has a Hilbert polynomial $p_{F} (t) = χ (X; F (t))$ . The polynomial is a numerical invariant whose leading coefficient encodes the dimension and degree of $X$ , and whose constant term encodes the arithmetic genus. The Castelnuovo-Mumford regularity refines this with bounds on when $H^{i} (F (t)) = 0$ .

Cox rings. The Cox ring of $P^{n}$ is $⨁_{d \geq 0} H^{0} (P^{n}; O (d)) = k [x_{0}, \dots, x_{n}]$ . For more general Mori-dream spaces and toric varieties, the Cox ring construction provides a uniform projective-coordinate description.

Toric perspective. $P^{n}$ is a toric variety — it admits an action by the torus $(k^{\times})^{n}$ with a dense open orbit. The combinatorial description: $P^{n}$ corresponds to the fan in $R^{n}$ generated by $e_{1}, e_{2}, \dots, e_{n}, - e_{1} - \dots - e_{n}$ . This generalises to toric varieties via fans and polytopes (Fulton Introduction to Toric Varieties).

Grassmannians and flag varieties. $P^{n} = Gr (1, n + 1)$ is the simplest Grassmannian. The general Grassmannian $Gr (k, n)$ parametrises $k$ -dimensional subspaces of $k^{n}$ and embeds in $P (Λ^{k} k^{n})$ via Plücker coordinates. Flag varieties $G / B$ for $G$ semisimple are the natural generalisation.

Brauer group and twisted projective forms. Over non-algebraically-closed fields, "projective space" admits Brauer-Severi twists: forms that become $P^{n}$ over the algebraic closure but are not $P^{n}$ over $k$ . The Brauer-Severi variety $X$ corresponds to a class in $Br (k)$ — central simple algebras over $k$ .

Crepant resolutions and $P^{n}$ as a target. Many natural moduli problems and singularities are resolved by maps to $P^{n}$ (Hirzebruch-Jung, weighted projective spaces, projective bundles).

Synthesis. This construction generalises the pattern fixed in 04.02.01 (scheme), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Detailed proofs of the cohomology of $O (d)$ (computing $H^{i}$ for all $i$ and $d$ ), the canonical bundle formula via the Euler sequence, the Picard group computation, the Hilbert polynomial degree-genus identification, and the Plücker relations are deferred to companion units. The key conceptual proofs are sketched in the formal-definition and exercises sections.

Connections [Master]

Scheme 04.02.01 — $P^{n}$ is the simplest non-affine scheme; $Proj k [x_{0}, \dots, x_{n}]$ .
Sheaf 04.01.01 — twisting sheaves $O (d)$ are foundational examples of coherent sheaves on $P^{n}$ .
Sheaf cohomology 04.03.01 — cohomology of $O (d)$ is the foundational computation.
Riemann-Roch theorem for curves 04.04.01 — projective curves embed in $P^{n}$ via line bundles; Riemann-Roch is computed by reduction to projective embedding.
Vector space 01.01.03 — $P^{n} = P (V)$ for $V = k^{n + 1}$ a vector space.
Projective scheme Proj(S) 04.02.03 — $P_{k}^{n} = Proj k [x_{0}, \dots, x_{n}]$ is the universal example.
Ample line bundle 04.05.05 — ampleness is defined relative to embeddings into $P^{n}$ .
Picard group 04.05.02 — $Pic (P_{k}^{n}) = Z$ is the canonical foundational example.

Historical & philosophical context [Master]

The geometric idea of projective space is older than the formalism. Renaissance painters (Alberti, Brunelleschi, ca. 1413–1450) discovered the vanishing point and developed the geometry of perspective. Gérard Desargues (1639) introduced Desargues's theorem and the synthetic projective plane, but the work was largely lost until rediscovered in the 19th century.

The modern algebraic formulation began with Hermann Grassmann's 1844 Die lineale Ausdehnungslehre — a remarkable but largely-unread work introducing what we now call exterior algebra and projective coordinates. August Möbius's 1827 Der barycentrische Calcul introduced homogeneous coordinates. Julius Plücker (1830s–60s) developed analytic projective geometry systematically, including the Plücker embedding of Grassmannians.

Arthur Cayley and Felix Klein in the late 19th century unified projective and Euclidean geometry through the Erlangen Programme: Euclidean geometry is the geometry preserved by a fixed quadric (the absolute) inside a projective space. David Hilbert's 1899 Grundlagen der Geometrie gave the modern axiomatic treatment.

The 20th century's algebraic and scheme-theoretic perspective is due to:

Severi, van der Waerden, Weil (1930s–40s): projective varieties as zero sets of homogeneous polynomial systems.
Cartan-Serre (FAC, 1955): the sheaf-cohomology computation of $H^{i} (P^{n}; O (d))$ was the prototype for the entire theory of coherent cohomology.
Grothendieck-Dieudonné (EGA, 1960s): the $Proj$ construction, projective morphisms, and the functorial characterisation. Grothendieck's relative perspective ( $P_{S}^{n}$ over an arbitrary base $S$ ) made projective space a universal moduli object.

Today projective space is the most-studied nonzero variety in mathematics, with deep ties to representation theory (flag varieties, Schubert calculus), number theory (heights, rational points, Mordell-Weil), physics (gauge theory's projective Higgs branches, twistor space in Penrose's programme), and combinatorics (matroid theory, tropical geometry). The single phrase "lines through the origin" generates an essentially endless mathematical landscape.

Bibliography [Master]

Hartshorne, Algebraic Geometry — §I.2, §II.4–§II.5 give the canonical scheme-theoretic treatment.
Vakil, The Rising Sea: Foundations of Algebraic Geometry — §4 covers Proj and projective schemes systematically.
Eisenbud & Harris, 3264 and All That — projective space as the home of intersection theory.
Grothendieck-Dieudonné, Éléments de Géométrie Algébrique II — the foundational reference for projective morphisms.
Mumford, The Red Book of Varieties and Schemes — classical introduction with strong projective focus.
Fulton, Introduction to Toric Varieties — projective space from the toric perspective.
Grassmann, Die lineale Ausdehnungslehre (1844) — the original synthetic foundation.
Möbius, Der barycentrische Calcul (1827) — homogeneous coordinates.
Plücker, Theorie der algebraischen Curven (1839) — analytic projective geometry.
Klein, "Vergleichende Betrachtungen über neuere geometrische Forschungen" (Erlangen Programme, 1872) — the philosophical foundation.

Prerequisites

04.02.01
01.01.03
01.01.01

Used in

04.05.05
04.07.02

Tier anchors

beginner: Strogatz analogous level for points-at-infinity completion of affine space
intermediate: Hartshorne §I.2, §II.4–§II.5; Vakil §4; Eisenbud-Harris *3264 and All That* §1
master: Hartshorne §II; Vakil; Grothendieck-Dieudonné EGA II; Eisenbud-Harris

References

TODO_REF
Hartshorne — Algebraic Geometry · §I.2, §II.4–§II.5, projective space and Proj
TODO_REF
Vakil — The Rising Sea · §4, projective schemes via Proj
TODO_REF
Grassmann — Die lineale Ausdehnungslehre · 1844, original synthetic projective geometry
TODO_REF
Grothendieck-Dieudonné — EGA II · Proj construction and projective morphisms
TODO_REF
Eisenbud-Harris — 3264 and All That · §1, projective space in intersection theory

Lean module

Codex.AlgebraicGeometry.Projective.ProjectiveSpace

Reviewer

TBD

Estimated time

beginner: 15m
intermediate: 45m
master: 65m