Projective geometry enlarges affine space so that parallel directions can meet at infinity and polynomial geometry behaves more uniformly under closure and intersection.
The affine guides Affine Varieties and Algebraic Sets and Ideals and Coordinate Rings worked inside Aⁿ. This guide adds one new coordinate, identifies scalar multiples, and obtains projective space Pⁿ.
That change looks small algebraically but transforms the global geometry. An affine parabola gains a point at infinity. Parallel affine lines meet on the line at infinity. Homogeneous equations become the correct global equations because scaling a projective coordinate vector must not change whether the equation vanishes.
Level: advanced undergraduate enrichment. We usually work over an algebraically closed field k when discussing classical projective varieties and intersection intuition.
Reading route: projective points → homogeneous coordinates → affine charts → homogeneous polynomials → projective varieties → projective closure → points at infinity → conics and intersections → practice → solutions.
1. Projective space is lines through the origin
Take kⁿ⁺¹\{0}. Declare two nonzero vectors equivalent when they differ by a nonzero scalar:
(x₀,…,xₙ)∼(λx₀,…,λxₙ), λ∈k*.
The set of equivalence classes is projective n-space Pⁿ(k).
A point is written [x₀:⋯:xₙ]. These are homogeneous coordinates. The brackets signal that the tuple is defined only up to common nonzero scale.
Geometrically, each projective point corresponds to a one-dimensional linear subspace through the origin in kⁿ⁺¹.
2. Homogeneous coordinates are not ordinary coordinates
[1:2:3]=[2:4:6]=[−1:−2:−3] whenever the scalars are allowed in k.
But [1:2:3]≠[1:2:4], because no single nonzero scalar transforms one vector into the other.
The zero tuple [0:⋯:0] is forbidden because it does not determine a line.
One cannot add projective points by adding coordinate representatives unless a separate geometric construction justifies it. Homogeneous coordinates are equivalence-class labels, not global affine vectors.
3. Affine space appears as a projective chart
Consider U₀={ [x₀:⋯:xₙ] : x₀≠0 }. Scale every point so x₀=1.
Then
[x₀:x₁:⋯:xₙ]=[1:x₁/x₀:⋯:xₙ/x₀].
Thus U₀≈Aⁿ by [1:a₁:⋯:aₙ]↔(a₁,…,aₙ).
The complement H∞={x₀=0} is itself a projective Pⁿ⁻¹ and is called the hyperplane at infinity relative to this affine chart.
Projective space is therefore affine space plus a structured family of directions at infinity, not one undifferentiated “point at infinity”.
4. The projective line is an affine line plus one point
P¹ has points [x₀:x₁]. On U₀ where x₀≠0, write [1:t], giving an affine coordinate t.
The only point outside U₀ has x₀=0, so it is [0:1]. Thus P¹=A¹∪{∞} set-theoretically in this chart.
A second chart U₁ uses coordinate s=x₀/x₁. On the overlap, s=1/t.
This reciprocal transition is why rational functions on the affine line naturally interact with the point at infinity.
5. Only homogeneous equations are well-defined projectively
A polynomial F(x₀,…,xₙ) is homogeneous of degree d if
F(λx₀,…,λxₙ)=λᵈF(x₀,…,xₙ).
If F is homogeneous, F(x)=0 is invariant under rescaling because λᵈ is nonzero when λ is nonzero.
If F is not homogeneous, vanishing may depend on the chosen representative and therefore fail to define a projective subset.
For example, x₀+x₁ is homogeneous of degree 1. But x₀+x₁² mixes degrees and does not define a projective equation without additional homogenisation.
MIT’s projective algebraic-geometry notes use exactly this definition of Pⁿ and Zariski-closed projective sets cut out by homogeneous polynomials: MIT 18.721 Projective Algebraic Geometry.
6. Projective algebraic sets are common zeros of homogeneous ideals
Let S be a family of homogeneous polynomials in k[x₀,…,xₙ]. Define V₊(S) as the set of projective points where every polynomial in S vanishes.
A homogeneous ideal is an ideal generated by homogeneous elements, equivalently one that contains every homogeneous component of each polynomial it contains.
Projective Zariski-closed sets correspond to homogeneous radical ideals, with the irrelevant ideal (x₀,…,xₙ) requiring special treatment because its affine zero set is only the forbidden origin.
The term “irrelevant ideal” is not a value judgement. It reflects that all positive-degree coordinates vanish only at the excluded zero vector.
7. Projective varieties use homogeneous prime ideals
Under the classical irreducible convention, a projective variety is an irreducible projective algebraic set.
Its homogeneous vanishing ideal is prime, again reflecting the irreducibility–prime correspondence.
For a projective hypersurface F=0 with homogeneous irreducible F, the principal ideal (F) is prime in many standard polynomial-ring settings and the hypersurface is irreducible.
As before, terminology varies. Some sources permit reducible “projective varieties”. Always check the convention before comparing statements.
8. Homogenisation adds the missing infinity terms
Suppose f(x₁,…,xₙ) has total degree d. Its homogenisation is
F(x₀,x₁,…,xₙ)=x₀ᵈ f(x₁/x₀,…,xₙ/x₀),
expanded so that F is a polynomial homogeneous of degree d.
Example: f(x,y)=y−x² has degree 2. Homogenising gives F(X,Y,Z)=YZ−X² if affine coordinates are x=X/Z, y=Y/Z.
Setting Z=1 recovers y−x²=0. Setting Z=0 reveals the points at infinity.
One must homogenise an ideal carefully. Homogenising a chosen generating set often works in simple examples, but in general the projective closure is controlled by the homogenised ideal and saturation, not naive term-by-term intuition alone.
9. Worked example: projective closure of the parabola
Take the affine parabola y=x² in A². In projective coordinates [X:Y:Z] with x=X/Z and y=Y/Z, its homogeneous equation is
YZ=X².
The affine chart Z≠0 recovers the original parabola.
At infinity set Z=0. Then X²=0, hence X=0. The remaining projective point is [0:1:0].
Thus the projective parabola has one point at infinity.
This point records the direction in which both ends of the affine parabola escape.
10. Parallel affine lines meet at infinity
Consider affine lines y=mx+b₁ and y=mx+b₂ with b₁≠b₂.
Homogenising in [X:Y:Z] gives Y=mX+b₁Z and Y=mX+b₂Z.
Subtracting gives (b₁−b₂)Z=0, so any projective intersection has Z=0.
Then Y=mX. The projective intersection point is [1:m:0], up to scaling.
Different slopes give different points on the line at infinity. Projective geometry therefore turns each affine direction into a point at infinity.
11. A projective conic is the natural global quadratic curve
A plane projective conic is cut out in P² by a homogeneous quadratic polynomial.
Examples include X²+Y²−Z²=0, XY−Z²=0 and YZ−X²=0.
Over an algebraically closed field of characteristic not two, every smooth projective conic is isomorphic to P¹ once it has a rational point over the field in question; over an algebraically closed field such a point exists.
Over non-algebraically-closed fields, rational-point existence becomes arithmetic. A conic may have geometric points over an extension but no point over the base field.
This is an early bridge from projective geometry to arithmetic geometry.
12. Worked parametrisation of a conic
Consider the projective conic XZ−Y²=0. Define
φ:P¹→P², [s:t]↦[s²:st:t²].
The image satisfies XZ−Y²=s²t²−s²t²=0.
The coordinate functions are homogeneous of the same degree 2, so the map is well-defined under rescaling [s:t].
This is the degree-two Veronese embedding of P¹ into P².
The example illustrates a general rule: projective maps are represented locally or globally by collections of homogeneous functions of equal degree with no simultaneous zero on the domain.
13. The projective plane is covered by three affine planes
P² has standard opens U_X={X≠0}, U_Y={Y≠0}, U_Z={Z≠0}. Each is isomorphic to A².
On U_Z, coordinates are x=X/Z and y=Y/Z. On U_X, coordinates may be u=Y/X and v=Z/X.
Projective calculations can therefore be reduced to affine calculations chart by chart.
A global projective variety may be nonsingular even if one chart equation looks awkward; another chart can reveal the local structure more clearly.
This chartwise philosophy leads naturally to schemes: global spaces are assembled from affine pieces with compatible coordinate rings.
14. Projective closure is not just “add Z and set Z=0” mechanically
For one hypersurface, homogenising its defining equation usually gives the correct closure directly.
For an ideal with several generators, naively homogenising those generators can introduce an unwanted component supported at infinity unless the homogenised ideal is saturated appropriately.
The robust principle is: embed Aⁿ into Pⁿ, take the Zariski closure of the image, and compute its homogeneous ideal.
Computationally, homogenisation plus saturation by the homogenising coordinate is a standard method.
This distinction matters because projective closure is a geometric operation, while a generator list is only one algebraic presentation.
15. Projective space behaves like a complete algebraic ambient space
Over C with the classical topology, Pⁿ(C) is compact. Algebraically, projective varieties satisfy a completeness/properness analogue: images of projective varieties under morphisms are closed under suitable hypotheses.
This is one reason projective geometry is central. Affine images can fail to be closed, while projective geometry often restores controlled global behaviour.
For example, the projection of the affine hyperbola xy=1 onto the x-axis has image A¹\{0}, not a closed affine subset. Proper/projective settings avoid this particular failure mode in the appropriate category.
The exact modern theorem is expressed using proper morphisms; projective morphisms are proper. The next cells move toward that language.
16. Bézout’s theorem explains why infinity improves intersection counts
For two projective plane curves of degrees m and n with no common component, Bézout’s theorem says their total intersection number over an algebraically closed field is mn when multiplicities are counted.
Affine curves can appear to “miss” intersections because some intersections occur at infinity.
Two distinct affine parallel lines have zero affine intersections, but their projective closures meet once at infinity, matching 1·1=1.
A line and conic generically meet in two projective points counting multiplicity. A tangent line contributes one geometric point with multiplicity two.
Intersection multiplicity is a deeper local algebraic notion; this guide uses Bézout only as structural motivation.
17. Degree is a projective invariant with intersection meaning
For a projective hypersurface, degree is simply the degree of its homogeneous defining polynomial.
For higher-codimension projective varieties, degree can be characterised by the number of intersection points with a generic complementary-dimensional linear subspace, counted with multiplicity.
A projective line has degree 1. A smooth conic has degree 2. The twisted cubic in P³ has degree 3.
Degree depends on the projective embedding. Isomorphic abstract varieties can admit different embeddings with different degrees.
This is another example of intrinsic versus embedded structure.
18. Rational maps may fail where all homogeneous coordinates vanish
A projective map can be given by homogeneous polynomials F₀,…,F_m of the same degree:
[x]↦[F₀(x):⋯:F_m(x)].
This is a morphism only where the Fᵢ do not vanish simultaneously.
If they have a common zero on the domain, the formula defines only a rational map with an indeterminacy locus.
Example: [X:Y:Z]↦[YZ:XZ:XY] is undefined at the three coordinate points because all outputs vanish there.
Birational geometry studies maps that are isomorphisms on dense open subsets even when global regularity fails.
19. Projective coordinate rings are graded
For a projective algebraic set X⊂Pⁿ with homogeneous ideal I(X), its homogeneous coordinate ring is
S(X)=k[x₀,…,xₙ]/I(X).
This ring inherits a grading by degree.
Unlike the affine coordinate ring, S(X) is not literally the ring of global regular functions on X. On a projective irreducible variety over an algebraically closed field, global regular functions are often only constants.
The graded structure instead controls the embedding and leads to Proj, Hilbert functions and projective schemes.
This distinction is essential before moving from affine Spec to projective Proj.
20. Hilbert functions measure graded growth
For a finitely generated graded coordinate ring S, the Hilbert function records dim_k S_d, the dimension of the degree-d homogeneous piece.
For large d, this function agrees with a polynomial, the Hilbert polynomial.
The degree of that polynomial gives the projective dimension, and its leading coefficient encodes the degree of the variety after standard normalisation.
This creates a bridge from counting monomials to geometric dimension and degree.
The later computational/enumerative R24 cell will return to Hilbert functions, Gröbner bases and intersection counting in more depth.
21. Common misconceptions
Projective space adds one point at infinity. Only P¹ does so relative to an affine chart. Aⁿ gains a whole Pⁿ⁻¹ hyperplane at infinity.
Any polynomial equation defines a projective set. It must be homogeneous, or converted into homogeneous equations.
Homogenising a generator list always gives the closure perfectly. Multiple generators may require saturation to remove spurious infinity components.
Homogeneous coordinate rings equal global function rings. Not in general. They encode the projective embedding through grading.
Projective means visually perspective drawing. Perspective geometry is a motivating special case; projective algebraic geometry is a much broader global polynomial theory.
22. Independent practice: eighteen questions
- Are [1:2:3] and [2:4:6] the same point?
- Why is [0:0:0] not a projective point?
- Identify the affine coordinates of [2:6:10] in U₀⊂P².
- What is the hyperplane at infinity for the chart Z≠0 in P²?
- Homogenise y−x².
- Find the point at infinity of the projective parabola YZ=X².
- Homogenise xy−1.
- Find the points at infinity of XY=Z².
- Where do affine parallel lines y=2x+1 and y=2x−3 meet projectively?
- Why must projective map coordinates have the same degree?
- Verify [s:t]↦[s²:st:t²] lands on XZ−Y²=0.
- What does Bézout predict for a line and conic with no common component?
- Why can a tangent line meet a conic at only one geometric point without contradicting Bézout?
- How many standard affine charts cover P²?
- What is wrong with [X:Y:Z]↦[YZ:XZ:XY] at [1:0:0]?
- What algebraic structure does the homogeneous coordinate ring remember that an ungraded ring presentation would hide?
- What does the degree of the Hilbert polynomial indicate?
- Why is the field relevant to whether a conic has a rational point?
23. Worked solutions and checks
1. Yes. The second vector is 2 times the first.
2. It spans no one-dimensional subspace and cannot be rescaled to a nonzero representative.
3. Divide by 2: [1:3:5], so affine coordinates are (3,5).
4. Z=0, a projective line P¹.
5. With x=X/Z and y=Y/Z, multiply by Z² to get YZ−X².
6. Set Z=0: X²=0, so X=0 and the unique point is [0:1:0].
7. xy−1 becomes XY−Z².
8. Set Z=0: XY=0. The points are [1:0:0] and [0:1:0].
9. Homogeneous equations are Y=2X+Z and Y=2X−3Z. Subtracting gives 4Z=0, so Z=0 and Y=2X. The point is [1:2:0].
10. Rescaling the input by λ should rescale every output coordinate by the same common factor so the projective image remains unchanged. Equal degree guarantees λᵈ appears in each coordinate.
11. XZ−Y²=s²t²−s²t²=0.
12. Two intersection points counting multiplicity over an algebraically closed field.
13. The single tangent intersection has multiplicity two.
14. Three: X≠0, Y≠0 and Z≠0.
15. All three output coordinates vanish, giving [0:0:0], which is not a projective point. The map is indeterminate there.
16. The degree grading of homogeneous pieces, which records how the variety sits in projective space.
17. Its degree equals the projective dimension.
18. A conic can acquire points only after extending the base field; geometric solvability and base-field rational solvability are different questions.
24. Where this guide hands off
The next R24 guide studies morphisms, tangent spaces, singularities and dimension. It will use affine charts to inspect projective points locally and compare Jacobian rank with intrinsic dimension.
Later R24 cells extend from classical varieties to schemes, sheaves, arithmetic geometry and computational/enumerative methods.
Sources and further study
See MIT 18.721 Projective Algebraic Geometry for projective varieties, homogeneous ideals, products and morphisms; MIT 18.721 Algebraic Geometry notes for the affine foundations; and MIT 18.782 lecture sequence for the transition from affine/projective varieties to curves and arithmetic applications.