Projective Varieties and Morphisms

Projective varieties are the zero loci of homogeneous polynomials in projective space

P^{n}

; morphisms are maps locally given by polynomial functions that respect their structure as ringed spaces. Key result: projective varieties are complete (the projective analogue of compactness), so the image of a projective variety under a morphism is closed — making projective geometry vastly better behaved than affine geometry. Where it applies: algebraic geometry, number theory, complex geometry, moduli theory, string theory. Who/when: classical foundations from Plücker, Cayley, and the Italian school (1830s–1930s); modern scheme-theoretic formulation by Serre and Grothendieck (1955–1965).

A projective variety over an algebraically closed field $k$ is a subset of projective space $P_{k}^{n}$ cut out by homogeneous polynomial equations, equipped with the Zariski topology and a sheaf of regular functions. The passage from affine to projective space adds "points at infinity," which compactifies the geometry and removes pathologies: parallel lines meet, the number of intersection points of curves becomes constant (Bézout), and the image of any morphism is closed. A morphism is a continuous map that pulls back regular functions to regular functions — concretely, away from base points it is given by tuples of homogeneous polynomials of equal degree. The surprising fact is that there are no nonconstant global regular functions on a connected projective variety (just as a holomorphic function on a compact complex manifold is constant), which forces all interesting geometry into sections of line bundles rather than functions.

The Core Insight

Projectivization trades the failure of affine geometry to be "complete" for the requirement that everything be graded and homogeneous. A point of $P^{n}$ is a line through the origin in $k^{n + 1}$ , so functions on $P^{n}$ must be insensitive to rescaling coordinates. The only scale-invariant polynomials are ratios of homogeneous polynomials of equal degree — these are the rational functions, and the regular ones glue to a sheaf. The deep payoff: $P^{n}$ is proper/complete, so projective morphisms are closed maps. This single property is the engine behind elimination theory, Bézout's theorem, intersection theory, and the entire edifice of moduli spaces.

Formal Statement

Projective Space

Let $k$ be a field. Projective $n$ -space is the set of one-dimensional linear subspaces of $k^{n + 1}$ :

P_{k}^{n} = (k^{n + 1} ∖ {0}) / \sim, (a_{0}, \dots, a_{n}) \sim (λ a_{0}, \dots, λ a_{n}) \forall λ \in k^{\times} .

⟳Two nonzero vectors span the same line iff they differ by a nonzero scalar; the equivalence classes are precisely those lines minus the origin.

The class of $(a_{0}, \dots, a_{n})$ is written $[a_{0} : \dots : a_{n}]$ (homogeneous coordinates). The subsets

U_{i} = {[a_{0} : \dots : a_{n}] : a_{i} \neq = 0}, P^{n} = i = 0 ⋃ n U_{i},

⟳At least one coordinate is nonzero in any class; the

U_{i}

cover

P^{n}

and each is in bijection with

A^{n}

via

[a_{0} : \dots : a_{n}] \mapsto (a_{0} / a_{i}, \dots, a_{i} / a_{i}, \dots, a_{n} / a_{i})

are the standard affine charts.

Homogeneous Ideals and Projective Varieties

A polynomial $f \in S = k [x_{0}, \dots, x_{n}]$ is homogeneous of degree $d$ if $f (λ x) = λ^{d} f (x)$ . While $f (a)$ is not well-defined on $P^{n}$ , the vanishing locus is, because $f (λa) = λ^{d} f (a) = 0 ⟺ f (a) = 0$ . An ideal $I \subseteq S$ is homogeneous if it is generated by homogeneous elements, equivalently $S / I$ inherits a grading $S = ⨁_{d \geq 0} S_{d}$ . The projective variety (or projective algebraic set) of $I$ is

V (I) = {[a] \in P^{n} : f (a) = 0 for all homogeneous f \in I} .

⟳Restrict attention to homogeneous generators; vanishing of each on a representative is independent of the chosen representative by the scaling identity above.

The closed sets of the Zariski topology on $P^{n}$ are exactly the sets $V (I)$ . A projective variety is irreducible if it is not the union of two proper closed subsets, equivalently if its homogeneous ideal $I (X)$ is prime.

Projective Nullstellensatz

Let $k = \overset{ˉ}{k}$ and let $m_{+} = (x_{0}, \dots, x_{n}) = ⨁_{d > 0} S_{d}$ be the irrelevant ideal. For a homogeneous ideal $I \subseteq S$ :

V (I) = \emptyset ⟺ I \supseteq m_{+}, and for V (I) \neq = \emptyset, I (V (I)) = I .

⟳Cone over

V (I)

A^{n + 1}

is the affine variety of

I

; apply the affine Nullstellensatz, noting the cone is

{0}

(or empty) iff

I \supseteq m_{+}

Thus there is an inclusion-reversing bijection between projective varieties in $P^{n}$ and radical homogeneous ideals properly contained in $m_{+}$ .

Regular Functions and Morphisms

Let $X \subseteq P^{n}$ be a projective variety with homogeneous coordinate ring $S (X) = S / I (X)$ . A function $φ : U \to k$ on an open $U \subseteq X$ is regular at $p$ if near $p$ it equals a ratio $g / h$ with $g, h \in S (X)$ homogeneous of the same degree and $h (p) \neq = 0$ . These form a sheaf $O_{X}$ , making $(X, O_{X})$ a ringed space.

A morphism $f : X \to Y$ of (quasi-)projective varieties is a continuous map such that for every open $V \subseteq Y$ and regular $ψ \in O_{Y} (V)$ , the pullback $ψ \circ f$ is regular on $f^{- 1} (V)$ :

f^{#} : O_{Y} (V) \to O_{X} (f^{- 1} (V)), f^{#} (ψ) = ψ \circ f .

⟳A morphism of varieties is exactly a morphism of locally ringed spaces; the condition that

f^{#}

lands in regular functions is the local model for "polynomial map" compatible with gluing across charts.

Concretely, a morphism $X \to P^{m}$ is given near each point by $[f_{0} : \dots : f_{m}]$ with the $f_{i} \in S (X)$ homogeneous of a common degree and not all vanishing.

Completeness / Properness of Projective Space — the Main Theorem of Elimination

For any variety $Z$ , the projection $π : P^{n} \times Z \to Z$ is a closed map: it sends Zariski-closed sets to Zariski-closed sets. Consequently every projective variety is complete, and:

f : X \to Y a morphism with X projective ⟹ f (X) \subseteq Y is closed.

⟳Reduce to eliminating projective coordinates from a system of homogeneous equations; the resultant/elimination ideal cuts out the (closed) image — this is classical elimination theory.

Why It Holds / Derivation

Completeness via Elimination

We prove $π : P^{n} \times A^{m} \to A^{m}$ is closed; the general case glues over affine charts of $Z$ .

Step 1 (Setup). A closed set

W \subseteq P^{n} \times A^{m}

is cut out by polynomials

g_{1}, \dots, g_{r}

x_{0}, \dots, x_{n}

(homogeneous in

x

) and

y_{1}, \dots, y_{m}

. For fixed

b \in A^{m}

, the fiber is

{[x] : g_{i} (x, b) = 0} \subseteq P^{n}

. Step 2 (Fiber emptiness condition). A point

b \in / π (W)

iff the system

g_{i} (x, b) = 0

has no solution in

P^{n}

. By the projective Nullstellensatz, this holds iff

m_{+}^{s} \subseteq (g_{1} (\cdot, b), \dots, g_{r} (\cdot, b)) for some s ≫ 0,

⟳No common projective zero

⟺

the irrelevant ideal lies in the radical

⟺

some power of every variable, hence

m_{+}^{s}

, lies in the ideal generated by the

g_{i} (\cdot, b)

Step 3 (Bound the degree). Let

S_{\geq s}

be polynomials of degree

\geq s

. Condition [7] says the

k

-linear map

μ_{b} : i ⨁ S_{s - d e g g_{i}} ⟶ S_{s}, (h_{i}) \mapsto i \sum h_{i} g_{i} (\cdot, b)

⟳

m_{+}^{s}

is spanned by all degree-

s

monomials, i.e.

S_{s}

; expressing each in the ideal is exactly surjectivity of

μ_{b}

in degree

s

is surjective for some

s

. Step 4 (Closedness of $π (W)$ ). Surjectivity of

μ_{b}

is the condition that certain maximal minors of the (polynomial-in-

b

) coefficient matrix do not all vanish. So

{b : μ_{b} surjective for this s}

is open, hence

A^{m} ∖ π (W) = ⋃_{s} {μ_{b} surjective at s}

is open and

π (W)

is closed.

■

No Nonconstant Global Regular Functions

Let $X$ be irreducible projective and $φ \in O_{X} (X)$ . By completeness, $φ$ defines a morphism $X \to A^{1} ↪ P^{1}$ whose image is closed and irreducible in $P^{1}$ , but the image misses the point $\infty$ , so it cannot be all of $P^{1}$ ; hence it is a single point. Therefore $φ$ is constant:

O_{X} (X) = k for X connected projective over k = \overset{ˉ}{k} .

⟳A proper irreducible closed subset of the irreducible curve

P^{1}

is a finite set of points; being the image of irreducible

X

it is irreducible, hence a single point.

■

Alexander Grothendieck | 1928–2014

Grothendieck recast projective varieties inside the theory of schemes, defining $Proj S$ for a graded ring $S$ so that classical projective varieties become $Proj k [x_{0}, \dots, x_{n}] / I$ . His relativization — projective morphisms $f : X \to S$ as closed subschemes of $P_{S}^{n}$ — turned "complete = proper" into a property checkable by the valuative criterion. Through EGA (1960–1967) he unified line bundles, cohomology (via $O (d)$ and Serre's theorems), and representability of functors, making projective geometry the cornerstone of moduli theory, where Hilbert and Quot schemes are themselves projective.

Building Intuition

Think of $P^{n}$ as $A^{n}$ plus a "horizon" $P^{n - 1}$ at infinity, one point for each direction. A homogeneous equation knows nothing about scale, so it describes a cone in $A^{n + 1}$ ; the projective variety is the cone "viewed from the origin." Two facts make projective geometry feel like compact complex geometry:

Completeness = compactness. Over $C$ , $P^{n} (C)$ is literally compact in the analytic topology. The algebraic "image is closed" theorem is the algebraic shadow of "the continuous image of a compact set is compact."

Functions are trivial; sections of bundles are everything. Just as a compact Riemann surface has no nonconstant holomorphic functions but rich spaces of meromorphic functions / line-bundle sections, a projective variety has $O_{X} (X) = k$ but rich graded pieces $S (X)_{d} = H^{0} (X, O_{X} (d))$ . All projective embeddings come from choosing such sections.

The mantra: affine = local/algebra of functions; projective = global/geometry of line bundles.

Affine charts covering $\mathbb{P}^1$ and the point at infinity; the conic $y^2=xz$ as a "horizon-completed" parabola.

Worked Examples

Bézout's Theorem on a Concrete Pair of Curves

Given: In

P^{2}

over

\overset{ˉ}{k}

, intersect a line

L = V (x_{0})

with the conic

C = V (x_{1}^{2} - x_{0} x_{2})

. Steps.

On $L$ we have $x_{0} = 0$ , so substitute: $C \cap L = V (x_{0}, x_{1}^{2}) \subset P^{2}$ .
The set-theoretic intersection is $[0 : 0 : 1]$ (need $x_{1} = 0$ , $x_{0} = 0$ , $x_{2} \neq = 0$ ): one point.
But Bézout's theorem counts with multiplicity = local intersection multiplicity. Here $x_{1}^{2}$ vanishes to order $2$ , so $I_{p} (C, L) = 2$ .
Bézout predicts $de g C \cdot de g L = 2 \cdot 1 = 2$ points counted with multiplicity.

# (C \cap L) = 2 (one point, multiplicity 2): tangency

Compactness of

P^{2}

is essential — in

A^{2}

moving the line off the curve could lose intersection points "at infinity"; projectively the count is invariant.

The Veronese Embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^2$

Given: The map

ν : P^{1} \to P^{2}

[s : t] \mapsto [s^{2} : s t : t^{2}]

, using degree-2 monomials. Steps.

Well-defined morphism: the three components are homogeneous of degree $2$ and never simultaneously zero (if all vanish then $s = t = 0$ , impossible). So $ν$ is a morphism by the criterion of equation [5].
Image is closed (by completeness): compute the relation among coordinates $[y_{0} : y_{1} : y_{2}] = [s^{2} : s t : t^{2}]$ . Then $y_{0} y_{2} = s^{2} t^{2} = (s t)^{2} = y_{1}^{2}$ .
So $ν (P^{1}) \subseteq V (y_{1}^{2} - y_{0} y_{2})$ , the conic — and one checks equality.
Isomorphism onto image: invert via $[y_{0} : y_{1} : y_{2}] \mapsto [y_{0} : y_{1}]$ where $y_{0} \neq = 0$ and $[y_{1} : y_{2}]$ where $y_{2} \neq = 0$ ; these agree on overlap, giving a regular inverse.

P^{1} \sim V (y_{1}^{2} - y_{0} y_{2}) \subset P^{2}, de g = 2

The smooth conic is projectively equivalent to

P^{1}

— every smooth conic is a rational curve.

Common Misconceptions

"Homogeneous polynomials define functions on $P^{n}$ ." They do not —

f (a)

depends on the chosen representative since

f (λa) = λ^{d} f (a)

. Only the vanishing locus is well-defined, and only ratios of equal-degree homogeneous polynomials give honest functions. The coordinate ring

S (X)

is not the ring of global functions; that ring is just

k

. The grading carries all the geometry. "Quasi-projective = projective minus a finite set." A quasi-projective variety is any locally closed subset (open in its closure) of

P^{n}

— including all affine varieties, which are not projective unless they are finite. Affine

A^{n}

is quasi-projective but not complete; the inclusion

A^{1} ↪ P^{1}

shows a morphism from a non-complete variety can have non-closed image. A morphism from an affine variety with non-closed image. Consider

f : A^{2} \to A^{2}

(x, y) \mapsto (x, x y)

. Its image is

{(a, b) : a \neq = 0} \cup {(0, 0)}

, which is not closed (it omits the punctured axis

{(0, b) : b \neq = 0}

but contains the origin). This pathology is impossible if the source is projective — completeness [6] forces closed images. This is the precise sense in which projective varieties are "better behaved." Not every variety is projective. Hironaka constructed a smooth complete

3

-fold that is not projective — completeness does not imply projectivity. In dimension

\leq 2

every smooth complete variety is projective, but in higher dimensions properness is strictly weaker than admitting an ample line bundle (embeddability into some

P^{N}

Connections

Projective varieties sit at the crossroads of several theories. Their structure as ringed spaces is the prototype for Schemes, where $Proj$ of a graded ring generalizes them. The grading $S (X) = ⨁_{d} S (X)_{d}$ realizes $S (X)_{d} = H^{0} (X, O_{X} (d))$ , linking to Line Bundles and Divisors and Sheaf Cohomology — Serre's vanishing $H^{i} (X, O_{X} (d)) = 0$ for $i > 0, d ≫ 0$ underlies ampleness. Over $C$ , GAGA and Analytification shows projective varieties coincide with compact complex analytic projective subvarieties, bridging to complex geometry. Intersection counts formalize via Bézout's Theorem and Intersection Theory, and the universal families of projective subvarieties form the projective Hilbert Scheme. The "no nonconstant functions" phenomenon parallels the maximum principle in Compact Riemann Surfaces.

Real-World Applications

Elliptic-curve cryptography (ECC). The group law on an elliptic curve $E \subset P^{2}$ uses the point at infinity $[0 : 1 : 0]$ as identity; projective coordinates avoid field inversions, accelerating ECDSA/X25519 in TLS and blockchain signatures.

Computer vision / multiple-view geometry. A pinhole camera is a projective morphism $P^{3} ⇢ P^{2}$ ; the fundamental matrix and homographies are projective transformations used in 3D reconstruction and SLAM.

Error-correcting codes. Projective Reed–Muller and algebraic-geometry (Goppa) codes are built from $H^{0} (X, O_{X} (d))$ on projective curves; they meet the Tsfasman–Vlăduț–Zink bound beating Gilbert–Varshamov.

Robotics and kinematics. Configuration spaces of mechanisms are real projective/algebraic varieties; solving inverse kinematics is elimination theory (resultants) on projective systems.

Number theory & Diophantine geometry. Rational points on projective varieties (the Mordell conjecture / Faltings' theorem) underpin modern arithmetic; projectivization is what makes "points at infinity" of Diophantine equations tractable.

String theory / mirror symmetry. Calabi–Yau hypersurfaces in weighted projective spaces and toric varieties are compactification manifolds; their intersection numbers feed Gromov–Witten invariants.

Geometric modeling (CAGD). Rational Bézier and NURBS curves/surfaces are images of polynomial maps into projective space, where weights are homogeneous coordinates.

Historical Development

1822 — Poncelet systematizes projective geometry and the principle of continuity, introducing points at infinity in his Traité des propriétés projectives.
1830s — Plücker, Möbius introduce homogeneous coordinates, making projective space computable.
1843 — Cayley & Salmon discover the 27 lines on a cubic surface, an early triumph of projective enumerative geometry.
1848 — Bézout's theorem rigorously framed projectively; intersection numbers become invariant only after compactification.
1860s–1900s — Riemann, Clebsch, Brill–Noether develop the function-theoretic side; the Italian school (Castelnuovo, Enriques, Severi) classifies projective surfaces.
1893 — Hilbert's Nullstellensatz gives the dictionary between ideals and varieties; the homogeneous version follows.
1900–1930 — van der Waerden, Zariski, Weil put foundations on rigorous algebraic footing; Zariski introduces the topology and normalization.
1955 — Serre's FAC ("Faisceaux algébriques cohérents") recasts projective varieties via coherent sheaves and cohomology, proving finiteness and ampleness theorems.
1956 — Serre's GAGA proves algebraic and analytic coherent sheaves agree on projective varieties over $C$ .
1960–67 — Grothendieck's EGA defines $Proj$ , projective morphisms, properness via the valuative criterion, and Hilbert/Quot schemes, completing the modern theory.

Subtleties & Limits

Several deep issues persist at the boundary of the theory:

Hartshorne's conjecture on complete intersections: every smooth subvariety of $P^{n}$ of dimension $> \frac{2}{3} n$ is conjecturally a complete intersection — open in general.
Effective Nullstellensatz / Bézout bounds: sharp degree bounds for the $h_{i}$ expressing $1 \in (f_{1}, \dots, f_{r})$ (and projective analogues) drive complexity in computer algebra; effective forms (Kollár, Brownawell) exist but optimal constants are subtle.
Properness vs. projectivity: characterizing exactly which complete varieties are projective (existence of an ample line bundle) is delicate; Hironaka's examples show the gap, and the question is governed by the structure of the Néron–Severi group and Kleiman's ampleness criterion.
Rational points: for projective varieties over $Q$ , deciding the existence/density of rational points (Bombieri–Lang, abc-influenced conjectures) remains largely open beyond curves.
Resolution & minimal models in positive characteristic affect when projective compactifications behave well.