Affine algebraic geometry begins with a simple-looking question: what geometric object is cut out by a collection of polynomial equations?
The answer is richer than the familiar graphs of school algebra. A polynomial equation can define a curve, surface, collection of components, isolated points or an empty set. Several equations can interact. Different equations can define exactly the same geometric set. A set that appears complicated in coordinates may be controlled by a remarkably small algebraic structure.
This guide develops the first cell of the BTT algebraic-geometry route. It assumes familiarity with polynomial algebra, sets, functions and basic proof. The topology background from Topological Spaces and Continuity is useful because algebraic geometry uses a topology very different from the Euclidean one.
Level: advanced undergraduate enrichment. The base field will usually be written k. Many classical statements simplify when k is algebraically closed, such as k=C. We will state when that hypothesis matters.
Reading route: polynomial zero sets → affine space → Zariski topology → irreducibility → varieties → examples → products and maps → dimension intuition → practice → solutions.
1. Affine space is the ambient coordinate world
The n-dimensional affine space over a field k is denoted Aⁿ(k), or simply Aⁿ when the field is understood. As a set it is kⁿ.
The point (a₁,…,aₙ) is not merely a vector to be added and scaled. In algebraic geometry, Aⁿ is treated as a space on which polynomial functions k[x₁,…,xₙ] can be evaluated.
For example, in A² the polynomial f(x,y)=y−x² vanishes exactly on the parabola y=x². The polynomial g(x,y)=xy vanishes on the union of the coordinate axes. Already these two examples show that a single equation can describe one irreducible-looking curve or several geometric components.
Over R, geometric pictures are visually familiar. Over C, points have complex coordinates and cannot be drawn directly in a real plane, but the algebra often becomes cleaner because nonconstant single-variable polynomials split into linear factors.
2. Algebraic sets are common zero loci of polynomial families
Given a collection S⊆k[x₁,…,xₙ], define
V(S)={p∈Aⁿ : f(p)=0 for every f∈S}.
Such a set is called an affine algebraic set. If S={f₁,…,f_m}, we also write V(f₁,…,f_m).
Examples: V(x)= the hyperplane x=0; V(x,y) in A² is the origin; V(xy) is the union V(x)∪V(y); V(x²+y²−1) over R is the unit circle.
The same zero set can be defined by many families. V(x)=V(x²)=V(x³,x⁵). Therefore the geometric object cannot be identified with one chosen list of equations.
The natural algebraic object is the ideal generated by the equations. If I=(S), then V(S)=V(I), because every polynomial combination of elements of S also vanishes wherever all elements of S vanish.
3. Ideals appear because polynomial consequences matter
An ideal I in k[x₁,…,xₙ] is a set closed under addition and under multiplication by arbitrary polynomials from the ring.
If f and g vanish on a set X, then f+g vanishes on X. If f vanishes on X and h is any polynomial, then hf also vanishes on X. Thus the collection of all polynomials vanishing on X automatically forms an ideal.
Define the vanishing ideal
I(X)={f∈k[x₁,…,xₙ] : f(p)=0 for all p∈X}.
There is therefore a two-way process: ideals produce zero sets by V, and subsets produce ideals by I. The second article in this R24 batch will study this algebra–geometry dictionary in depth through coordinate rings and the Nullstellensatz.
4. Inclusion reverses between equations and zero sets
If S⊆T are polynomial families, then V(T)⊆V(S). More equations create more restrictions, so the zero set can only shrink.
Similarly, if X⊆Y are subsets of affine space, then I(Y)⊆I(X). A polynomial that vanishes on the larger set automatically vanishes on the smaller one.
This order reversal is fundamental. Algebraic geometry repeatedly converts geometric inclusion questions into ideal containments in the opposite direction.
For example, in A² we have V(x,y)⊆V(x), so I(V(x))⊆I(V(x,y)). The origin lies inside the y-axis, while the set of all polynomials vanishing on the y-axis is contained in the larger ideal of polynomials vanishing at the origin.
5. The Zariski topology declares algebraic sets closed
The Zariski topology on Aⁿ is defined by taking affine algebraic sets as the closed sets.
This really is a topology because: the empty set and whole affine space are algebraic; arbitrary intersections of algebraic sets are algebraic; finite unions are algebraic.
The finite-union identity comes from V(I)∪V(J)=V(IJ)=V(I∩J). In the simplest case V(f)∪V(g)=V(fg).
The Zariski topology is much coarser than the Euclidean topology over R or C. In A¹ over an infinite field, every proper algebraic set is finite, because a nonzero one-variable polynomial has only finitely many roots. Thus every nonempty Zariski-open subset of A¹ is the complement of a finite set.
This changes geometric intuition. Two nonempty Zariski-open subsets of A¹ necessarily intersect. Small Euclidean balls are not the basic local objects of algebraic geometry.
MIT’s undergraduate algebraic-geometry notes and Harvard Math 137 notes use this same progression from affine algebraic sets to the Zariski topology and coordinate rings: MIT 18.721 Algebraic Geometry notes and Harvard undergraduate algebraic geometry notes.
6. Worked example: why V(xy) is a union
Consider X=V(xy)⊂A². A point (a,b) lies in X exactly when ab=0. Over any field, this means a=0 or b=0 because fields have no zero divisors.
Therefore
V(xy)=V(x)∪V(y).
The set is the union of the y-axis and x-axis. It is reducible because it can be expressed as a union of two proper closed subsets.
Compare with V(y−x²). That parabola cannot be split into a union of two proper affine algebraic subsets over an algebraically closed field; its defining polynomial is irreducible, and the corresponding principal ideal is prime.
7. Irreducibility is the algebraic substitute for being one piece
A nonempty topological space X is irreducible if it cannot be written as X=Y∪Z with Y and Z proper closed subsets of X.
This is not the same as connectedness in ordinary topology. Irreducibility is stronger in the Zariski setting. An irreducible space is connected, but a connected space need not be irreducible.
For an affine algebraic set X over an algebraically closed field, X is irreducible exactly when I(X) is a prime ideal. This converts a geometric decomposition question into a ring-theoretic condition.
Why prime? If fg vanishes on X and X is irreducible, then X=V(f)∪V(g) inside X. Irreducibility forces X⊆V(f) or X⊆V(g), hence f∈I(X) or g∈I(X).
8. Varieties are irreducible algebraic spaces in the classical convention
Terminology varies between texts. In the classical affine setting, an affine variety often means an irreducible affine algebraic set. Some modern or computational texts use “variety” more loosely for reduced finite-type algebraic sets, possibly reducible.
This guide uses the classical convention: affine variety = irreducible affine algebraic set, unless another convention is explicitly stated.
The convention matters because the algebraic partner changes. Irreducible affine varieties correspond to prime ideals, while arbitrary affine algebraic sets correspond to radical ideals over an algebraically closed field.
When reading external sources, check the definition before comparing theorems. A disagreement may be terminological rather than mathematical.
9. Basic examples of affine varieties
Affine space Aⁿ. It is irreducible because the zero ideal in k[x₁,…,xₙ] is prime when k is a field.
Affine line. A¹ is the simplest positive-dimensional variety. Its proper closed subsets are finite over an algebraically closed field.
Parabola V(y−x²). The map t↦(t,t²) gives a polynomial parametrisation and in fact an isomorphism with A¹.
Hyperbola V(xy−1). Neither x nor y can vanish. Solving y=1/x shows that this is naturally related to the multiplicative group k*.
Cusp V(y²−x³). It is irreducible over many fields of interest but singular at the origin. The parametrisation t↦(t²,t³) is bijective over algebraically closed fields, yet its inverse is not regular at the cusp in the variety-theoretic sense. This anticipates the difference between set-theoretic bijection and isomorphism.
10. Worked example: the parabola is isomorphic to the affine line
Let X=V(y−x²)⊂A². Define φ:A¹→X by φ(t)=(t,t²). This is a polynomial map.
Define ψ:X→A¹ by ψ(x,y)=x. This is the restriction of the coordinate function x, hence regular.
Then ψ∘φ(t)=t, and φ∘ψ(x,y)=(x,x²)=(x,y) because points of X satisfy y=x².
Thus A¹ and the parabola are isomorphic as affine varieties. The geometric curve may be bent in the ambient plane, but algebraically it has the same intrinsic affine-variety structure as a line.
This is an early warning against confusing embedding shape with intrinsic algebraic structure.
11. Polynomial maps are the first morphisms
A polynomial map F:Aⁿ→Aᵐ has coordinate functions F=(f₁,…,f_m) with each fᵢ∈k[x₁,…,xₙ].
If X⊂Aⁿ and Y⊂Aᵐ are algebraic sets and F(X)⊂Y, then the restriction F|_X is a regular map in the simplest affine sense.
Regular maps are continuous for the Zariski topology. If Z=V(S) is closed in Y, then F⁻¹(Z) is cut out on X by the pulled-back polynomials f∘F.
The reverse viewpoint is algebraic: a regular map X→Y induces a homomorphism of coordinate rings in the opposite direction by composition. The contravariance of geometry and functions will be central in the next article.
12. Products remain algebraic
If X⊂Aⁿ and Y⊂Aᵐ are affine algebraic sets, then X×Y⊂Aⁿ⁺ᵐ is again algebraic.
If X=V(f₁,…,f_r) in x-variables and Y=V(g₁,…,g_s) in y-variables, then X×Y is cut out by the same equations viewed in k[x₁,…,xₙ,y₁,…,y_m].
The coordinate ring later satisfies k[X×Y]≈k[X]⊗_k k[Y] under standard hypotheses. This is an example of a geometric construction being mirrored by an algebraic construction.
Products also help distinguish intrinsic and ambient dimension. A¹×A¹ is A², and dimensions add. For irreducible affine varieties, dim(X×Y)=dimX+dimY.
13. Dimension begins with chains of irreducible closed sets
The dimension of an irreducible affine variety can be defined as the maximum length of a strict chain
X₀⊊X₁⊊⋯⊊X_d=X
of irreducible closed subsets. The integer d is the dimension when such a finite maximum exists, as it does for affine varieties of finite type over a field.
For A¹, a maximal chain is {point}⊊A¹, so dimension is 1. For A², one has {point}⊊{irreducible curve}⊊A², so dimension is 2.
Algebraically, the dimension of an affine variety agrees with the Krull dimension of its coordinate ring. It also equals the transcendence degree of its function field over k when the variety is irreducible.
MIT’s algebraic-geometry materials explicitly place dimension after affine and projective morphisms and integral extensions: MIT 18.721 course notes.
14. Equation count is only a heuristic for dimension
It is tempting to say “n variables minus r equations gives dimension n−r”. This is sometimes correct when the equations impose independent conditions, but it is not a universal rule.
For example, in A² the equations x=0 and x²=0 impose the same geometric condition. They do not reduce dimension twice.
The equations x=0 and xy=0 also do not impose independent restrictions once x=0 is known. Their common zero set is still the y-axis, dimension 1.
In a smooth complete-intersection setting, independent equations often reduce local dimension by their rank. Singularities and algebraic dependence can cause the naive count to fail.
The fourth article in this batch will use tangent spaces and Jacobian rank to make that local dimension heuristic precise under suitable hypotheses.
15. Components separate a reducible algebraic set into irreducible pieces
Every affine algebraic set over a Noetherian ring setting decomposes into finitely many irreducible components, none contained in another.
For V(xy) the components are V(x) and V(y). For V((y−x²)(y+x²)) over a field of characteristic not two, the set is the union of the two curves y=x² and y=−x².
Algebraically, irreducible components correspond to minimal prime ideals over the defining radical ideal.
This shows why factorisation matters geometrically. A product equation can encode a union. But one should factor the ideal structure, not merely one convenient generator, because several equations can interact in ways that are not visible from one polynomial.
16. The field changes the geometry
Consider V(x²+y²) in A².
Over R, the only point is (0,0). Over C, the polynomial factors as (x+iy)(x−iy), so the zero set is the union of two complex lines.
Thus algebraic geometry cannot be separated from the choice of base field. Statements about irreducibility, number of points and factorisation may change after extending the field.
A polynomial irreducible over R may become reducible over C. Conversely, arithmetic geometry studies how varieties behave over fields such as Q or finite fields where rational points become a central problem.
This is why advanced algebraic geometry distinguishes geometric irreducibility from irreducibility over the base field.
17. The Zariski closure records all polynomial consequences
For any subset S⊂Aⁿ, its Zariski closure is V(I(S)). It is the smallest algebraic set containing S.
If S is infinite in A¹ over an algebraically closed field, then no nonzero polynomial can vanish on all of S unless the set happens to lie among finitely many roots, which is impossible. Hence I(S)=(0) and the Zariski closure is all of A¹.
This is dramatically different from Euclidean closure. The set of positive integers inside C is discrete in the ordinary topology but Zariski-dense in A¹(C).
Density therefore means “no nonzero polynomial equation cuts the set down further”, not “every small Euclidean neighbourhood contains a point of the set”.
18. Parametrisation can hide algebraic relations
Suppose φ:A¹→A² is given by t↦(t²,t³). The image lies in V(y²−x³) because (t³)²−(t²)³=0.
Eliminating the parameter t reveals the polynomial relation y²=x³. This is an elementary instance of elimination theory.
In general, the image of a polynomial map need not be closed in the Zariski topology, but its closure is algebraic. Elimination ideals and Gröbner bases provide computational methods for finding equations satisfied by the image.
This begins the computational side of algebraic geometry: geometry asks for the locus; commutative algebra provides exact operations for discovering its defining relations.
19. A dependable affine-geometry workflow
First state the base field. Then state the ambient affine space and the defining polynomial family or ideal.
Check whether different equations are redundant. Factor when useful, but remember that factorisation of one generator may not reveal the full ideal decomposition.
Separate the set from its chosen equations. Use I(X) when the complete family of polynomial consequences matters.
When claiming irreducibility, either give a geometric argument or prove the corresponding ideal is prime. When claiming a decomposition, verify that the proposed components are closed and proper.
When estimating dimension, treat “variables minus equations” as a heuristic until independence, prime height, transcendence degree or tangent-space/Jacobian evidence supports it.
20. Common misconceptions
Misconception 1: the defining equations are unique. They are not. The geometry is attached to the zero set or its vanishing ideal, not one presentation.
Misconception 2: Zariski-open means a tiny neighbourhood. It usually does not. Zariski opens can be very large in Euclidean terms.
Misconception 3: connected means irreducible. Irreducibility is stronger and algebraically tied to prime ideals.
Misconception 4: one polynomial equation always reduces dimension by one. Only under appropriate nondegeneracy or height assumptions.
Misconception 5: a bijective polynomial map must be an isomorphism. False in general; the inverse must also be regular. Singular examples expose the distinction.
21. Independent practice: sixteen questions
- Describe V(xy) in A².
- Describe V(x,y) in A².
- Show V(f)∪V(g)=V(fg).
- If I⊆J, compare V(I) and V(J).
- Find I({(0,0)}) inside k[x,y].
- Over an infinite field, why is every proper closed subset of A¹ finite?
- Show V(y−x²) is isomorphic to A¹.
- Explain why V(xy) is reducible.
- Over C, decompose V(x²+y²).
- Over R, what is V(x²+y²)?
- Find a polynomial relation satisfied by the image of t↦(t²,t³).
- Why does x=0 together with x²=0 not cut A² down by two dimensions?
- Give a maximal chain of irreducible closed subsets showing dim A²≥2.
- Why is an infinite subset of A¹ over an algebraically closed field Zariski-dense?
- What algebraic property corresponds to irreducibility of an affine algebraic set over an algebraically closed field?
- Explain one reason the choice of base field matters.
22. Worked solutions and checks
1. xy=0 means x=0 or y=0, so V(xy)=V(x)∪V(y), the union of the coordinate axes.
2. Both coordinates must vanish, so V(x,y)={(0,0)}.
3. A point satisfies fg=0 exactly when f=0 or g=0 because field values have no zero divisors. Hence the zero set of fg is the union.
4. More equations mean fewer points, so V(J)⊆V(I).
5. A polynomial vanishes at the origin exactly when its constant term is zero. Therefore I({0})=(x,y).
6. A proper closed set is V(S) for some family containing a nonzero polynomial f. Then it lies in V(f), which has finitely many roots in one variable.
7. Use φ(t)=(t,t²) and ψ(x,y)=x on the parabola. Their compositions are the identities.
8. It is the union of the two proper closed sets V(x) and V(y).
9. x²+y²=(x+iy)(x−iy), so the set is V(x+iy)∪V(x−iy).
10. x²+y²=0 over R forces x=y=0, so the set is the origin.
11. y²−x³=0 because t⁶−t⁶=0.
12. The second equation is already a polynomial consequence of the first. It adds no new geometric restriction.
13. {(0,0)}⊊V(y)⊊A² is a strict chain of irreducible closed subsets, so dimension is at least two. Standard theory shows it is exactly two.
14. A nonzero one-variable polynomial has only finitely many roots. Therefore no nonzero polynomial vanishes on an infinite subset, so its vanishing ideal is zero and its closure is all A¹.
15. The vanishing ideal is prime.
16. Factorisation and point sets can change after extending the field; x²+y² is the standard example over R versus C.
23. Where this guide hands off
The next R24 cell turns the geometry into algebra. It studies coordinate rings k[X], radical and prime ideals, the Nullstellensatz, regular functions and the reversal between geometric maps and ring homomorphisms.
After that, projective geometry adds points at infinity and homogeneous coordinates, and the fourth guide develops morphisms, tangent spaces, singularities and dimension.
Sources and further study
See MIT 18.721 Algebraic Geometry notes for affine algebraic geometry, Zariski topology, spectra, localisation and morphisms; Harvard Math 137 undergraduate algebraic geometry notes for affine varieties, coordinate rings and regular maps; and MIT 18.782 lecture sequence for an arithmetic route that transitions from affine and projective varieties into curves and number-theoretic applications.