Projective Varieties and Morphisms

⚡ Key Takeaways

Projective varieties are the zero loci of homogeneous polynomials in projective space $\mathbb{P}^n$ ; 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).

💡 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 $\mathbb{P}^n$ is a line through the origin in $k^{n+1}$ , so functions on $\mathbb{P}^n$ 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: $\mathbb{P}^n$ 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.

DefinitionProjective Space

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

1

\quad \mathbb{P}^n_k = \big(k^{n+1}\setminus\{0\}\big)\big/\sim, \qquad (a_0,\dots,a_n)\sim(\lambda a_0,\dots,\lambda a_n)\ \forall\,\lambda\in k^\times.

Derivation hint: 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 $[a_0:\cdots:a_n]$ (homogeneous coordinates). The subsets

2

\quad U_i = \{[a_0:\cdots:a_n] : a_i \neq 0\}, \qquad \mathbb{P}^n = \bigcup_{i=0}^n U_i,

Derivation hint: At least one coordinate is nonzero in any class; the $U_i$ cover $\mathbb{P}^n$ and each is in bijection with $\mathbb{A}^n$ via $[a_0:\cdots:a_n]\mapsto(a_0/a_i,\dots,\widehat{a_i/a_i},\dots,a_n/a_i)$ . are the standard affine charts.

DefinitionHomogeneous Ideals and Projective Varieties

A polynomial $f\in S = k[x_0,\dots,x_n]$ is homogeneous of degree $d$ if $f(\lambda x) = \lambda^d f(x)$ . While $f(a)$ is not well-defined on $\mathbb{P}^n$ , the vanishing locus is, because $f(\lambda a)=\lambda^d f(a)=0 \iff f(a)=0$ . An ideal $I\subseteq S$ is homogeneous if it is generated by homogeneous elements, equivalently $S/I$ inherits a grading $S = \bigoplus_{d\geq 0} S_d$ . The projective variety (or projective algebraic set) of $I$ is

3

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

Derivation hint: 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 $\mathbb{P}^n$ 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.

TheoremProjective Nullstellensatz

Let $k = \bar k$ and let $\mathfrak{m}_+ = (x_0,\dots,x_n) = \bigoplus_{d>0} S_d$ be the irrelevant ideal. For a homogeneous ideal $I\subseteq S$ :

4

\quad V(I) = \varnothing \iff \sqrt{I} \supseteq \mathfrak{m}_+, \qquad\text{and for } V(I)\neq\varnothing,\ \ I\big(V(I)\big) = \sqrt{I}.

Derivation hint: Cone over $V(I)$ in $\mathbb{A}^{n+1}$ is the affine variety of $I$ ; apply the affine Nullstellensatz, noting the cone is $\{0\}$ (or empty) iff $\sqrt I\supseteq\mathfrak m_+$ . Thus there is an inclusion-reversing bijection between projective varieties in $\mathbb{P}^n$ and radical homogeneous ideals properly contained in $\mathfrak m_+$ .

DefinitionRegular Functions and Morphisms

Let $X\subseteq\mathbb{P}^n$ be a projective variety with homogeneous coordinate ring $S(X)=S/I(X)$ . A function $\varphi:U\to k$ 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 $\mathcal{O}_X$ , making $(X,\mathcal{O}_X)$ 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 $\psi\in\mathcal{O}_Y(V)$ , the pullback $\psi\circ f$ is regular on $f^{-1}(V)$ :

5

\quad f^\# : \mathcal{O}_Y(V) \to \mathcal{O}_X\big(f^{-1}(V)\big), \qquad f^\#(\psi) = \psi\circ f.

Derivation hint: 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 $X\to\mathbb{P}^m$ is given near each point by $[f_0:\cdots:f_m]$ with the $f_i\in S(X)$ homogeneous of a common degree and not all vanishing.

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

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

6

\quad f:X\to Y \text{ a morphism with } X \text{ projective} \ \Longrightarrow\ f(X)\subseteq Y \text{ is closed.}

Derivation hint: 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.

ProofCompleteness via Elimination

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

Step 1 (Setup). A closed set $W\subseteq\mathbb{P}^n\times\mathbb{A}^m$ is cut out by polynomials $g_1,\dots,g_r$ in $x_0,\dots,x_n$ (homogeneous in $x$ ) and $y_1,\dots,y_m$ . For fixed $b\in\mathbb{A}^m$ , the fiber is $\{[x]: g_i(x,b)=0\}\subseteq\mathbb{P}^n$ .

Step 2 (Fiber emptiness condition). A point $b\notin\pi(W)$ iff the system $g_i(x,b)=0$ has no solution in $\mathbb{P}^n$ . By the projective Nullstellensatz, this holds iff

7

\quad \mathfrak{m}_+^{\,s} \subseteq \big(g_1(\cdot,b),\dots,g_r(\cdot,b)\big) \quad \text{for some } s\gg 0,

Derivation hint: No common projective zero $\iff$ the irrelevant ideal lies in the radical $\iff$ some power of every variable, hence $\mathfrak m_+^s$ , 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

8

\quad \mu_b:\bigoplus_{i} S_{s-\deg g_i} \longrightarrow S_s, \qquad (h_i)\mapsto \sum_i h_i\, g_i(\cdot,b)

Derivation hint: $\mathfrak m_+^s$ is spanned by all degree- $s$ monomials, i.e. $S_s$ ; expressing each in the ideal is exactly surjectivity of $\mu_b$ in degree $s$ . is surjective for some $s$ .

Step 4 (Closedness of $\pi(W)$ ). Surjectivity of $\mu_b$ is the condition that certain maximal minors of the (polynomial-in- $b$ ) coefficient matrix do not all vanish. So $\{b: \mu_b \text{ surjective for this } s\}$ is open, hence $\mathbb{A}^m\setminus\pi(W) = \bigcup_s\{\mu_b \text{ surjective at } s\}$ is open and $\pi(W)$ is closed. $\blacksquare$

∎

ProofNo Nonconstant Global Regular Functions

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

9

\quad \mathcal{O}_X(X) = k \quad\text{for } X \text{ connected projective over } k=\bar k.

Derivation hint: A proper irreducible closed subset of the irreducible curve $\mathbb{P}^1$ is a finite set of points; being the image of irreducible $X$ it is irreducible, hence a single point. $\blacksquare$

∎

mathematician · Alexander Grothendieck | 1928–2014

Grothendieck recast projective varieties inside the theory of schemes, defining $\operatorname{Proj} S$ for a graded ring $S$ so that classical projective varieties become $\operatorname{Proj} k[x_0,\dots,x_n]/I$ . His relativization — projective morphisms $f:X\to S$ as closed subschemes of $\mathbb{P}^n_S$ — turned "complete = proper" into a property checkable by the valuative criterion. Through EGA (1960–1967) he unified line bundles, cohomology (via $\mathcal{O}(d)$ and Serre's theorems), and representability of functors, making projective geometry the cornerstone of moduli theory, where Hilbert and Quot schemes are themselves projective.

🧠 Intuition

Think of $\mathbb{P}^n$ as $\mathbb{A}^n$ plus a "horizon" $\mathbb{P}^{n-1}$ at infinity, one point for each direction. A homogeneous equation knows nothing about scale, so it describes a cone in $\mathbb{A}^{n+1}$ ; the projective variety is the cone "viewed from the origin." Two facts make projective geometry feel like compact complex geometry: 1. Completeness = compactness. Over $\mathbb{C}$ , $\mathbb{P}^n(\mathbb{C})$ 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." 2. 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 $\mathcal{O}_X(X)=k$ but rich graded pieces $S(X)_d = H^0(X,\mathcal{O}_X(d))$ . All projective embeddings come from choosing such sections. The mantra: affine = local/algebra of functions; projective = global/geometry of line bundles.

Figure

Affine charts covering

\mathbb{P}^1

and the point at infinity; the conic

y^2=xz

as a "horizon-completed" parabola.

ExampleBézout's Theorem on a Concrete Pair of Curves

Given: In $\mathbb{P}^2$ over $\bar k$ , intersect a line $L = V(x_0)$ with the conic $C = V(x_1^2 - x_0 x_2)$ .

Steps. 1. On $L$ we have $x_0 = 0$ , so substitute: $C\cap L = V(x_0,\, x_1^2) \subset \mathbb{P}^2$ . 2. The set-theoretic intersection is $[0:0:1]$ (need $x_1=0$ , $x_0=0$ , $x_2\neq 0$ ): one point. 3. But Bézout's theorem counts with multiplicity = local intersection multiplicity. Here $x_1^2$ vanishes to order $2$ , so $I_p(C,L)=2$ . 4. Bézout predicts $\deg C\cdot\deg L = 2\cdot 1 = 2$ points counted with multiplicity.

\boxed{\#(C\cap L) = 2 \text{ (one point, multiplicity } 2\text{): tangency}}

Compactness of $\mathbb{P}^2$ is essential — in $\mathbb{A}^2$ moving the line off the curve could lose intersection points "at infinity"; projectively the count is invariant.

ExampleThe Veronese Embedding

\mathbb{P}^1\hookrightarrow\mathbb{P}^2

Given: The map $\nu:\mathbb{P}^1\to\mathbb{P}^2$ , $[s:t]\mapsto[s^2:st:t^2]$ , using degree-2 monomials.

Steps. 1. Well-defined morphism: the three components are homogeneous of degree $2$ and never simultaneously zero (if all vanish then $s=t=0$ , impossible). So $\nu$ is a morphism by the criterion of equation [5]. 2. Image is closed (by completeness): compute the relation among coordinates $[y_0:y_1:y_2]=[s^2:st:t^2]$ . Then $y_0 y_2 = s^2 t^2 = (st)^2 = y_1^2$ . 3. So $\nu(\mathbb{P}^1) \subseteq V(y_1^2 - y_0 y_2)$ , the conic — and one checks equality. 4. 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.

\boxed{\mathbb{P}^1 \;\xrightarrow{\ \sim\ }\; V(y_1^2 - y_0 y_2)\subset\mathbb{P}^2, \quad \deg = 2}

The smooth conic is projectively equivalent to $\mathbb{P}^1$ — every smooth conic is a rational curve.

⚠️ Common mistake

"Homogeneous polynomials define functions on $\mathbb{P}^n$ ." They do not — $f(a)$ depends on the chosen representative since $f(\lambda a)=\lambda^d f(a)$ . 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.

⚠️ Common mistake

"Quasi-projective = projective minus a finite set." A quasi-projective variety is any locally closed subset (open in its closure) of $\mathbb{P}^n$ — including all affine varieties, which are not projective unless they are finite. Affine $\mathbb{A}^n$ is quasi-projective but not complete; the inclusion $\mathbb{A}^1\hookrightarrow\mathbb{P}^1$ shows a morphism from a non-complete variety can have non-closed image.

Counterexample

A morphism from an affine variety with non-closed image. Consider $f:\mathbb{A}^2\to\mathbb{A}^2$ , $(x,y)\mapsto(x, xy)$ . 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."

Counterexample

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 $\mathbb{P}^N$ ).

connection

Projective varieties sit at the crossroads of several theories. Their structure as ringed spaces is the prototype for Schemes, where $\operatorname{Proj}$ of a graded ring generalizes them. The grading $S(X)=\bigoplus_d S(X)_d$ realizes $S(X)_d = H^0(X,\mathcal{O}_X(d))$ , linking to Line Bundles and Divisors and Sheaf Cohomology — Serre's vanishing $H^i(X,\mathcal{O}_X(d))=0$ for $i>0,\,d\gg0$ underlies ampleness. Over $\mathbb{C}$ , 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.

🔬 Where this appears

⏳ History

- 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

\mathbb{C}

.

- 1960–67 — Grothendieck's EGA defines

\operatorname{Proj}

, projective morphisms, properness via the valuative criterion, and Hilbert/Quot schemes, completing the modern theory.

open-problem

Several deep issues persist at the boundary of the theory:

Hartshorne's conjecture on complete intersections: every smooth subvariety of $\mathbb{P}^n$ of dimension $> \tfrac{2}{3}n$ 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 $\mathbb{Q}$ , 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.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also