Algebraic geometry becomes local when we stop asking only which points satisfy the equations and begin asking what the space looks like infinitesimally near each point.
The first three R24 guides built the global language: affine varieties, coordinate rings, and projective varieties. This guide adds the local tests that distinguish a smooth algebraic point from a singular one and connect the visible equations to intrinsic dimension.
A singularity is not merely a sharp-looking picture. It is an algebraic failure of the local space to look regular in the expected dimension. A cusp, a crossing, a cone vertex and a nonreduced thickening can all be singular for different reasons. The tangent space, local ring and Jacobian provide complementary ways to see the failure.
Level: advanced undergraduate enrichment. For the smoothness calculations below, assume an algebraically closed field of characteristic zero unless stated otherwise. The definitions using local rings make sense much more generally; positive-characteristic smoothness needs additional care.
Reading route: morphisms → pullback → rational maps → local rings → tangent spaces → Jacobians → singular examples → dimension and codimension → fibres → normalisation → projective checks → practice → solutions.
1. Morphisms are the structure-preserving maps of algebraic geometry
For affine varieties X⊂Aⁿ and Y⊂Aᵐ, a morphism φ:X→Y is given by regular coordinate functions:
φ(p)=(f₁(p),…,f_m(p)),
where each fᵢ is regular on X and the resulting point satisfies the defining equations of Y.
In the simplest affine setting the fᵢ are classes in k[X]. On principal opens they may be represented by fractions whose denominators do not vanish.
Morphisms compose, identity maps are morphisms, and an isomorphism is a morphism with a morphism inverse.
A continuous bijection for the Zariski topology is not automatically an isomorphism. Algebraic geometry insists that the inverse preserve regular functions as well.
2. Pullback turns geometry into algebra in the opposite direction
A morphism φ:X→Y induces a k-algebra homomorphism
φ*:k[Y]→k[X], f↦f∘φ.
The direction reverses because a function on the target can be observed on X by first applying φ.
Conversely, for affine algebraic sets over an algebraically closed field, a k-algebra homomorphism k[Y]→k[X] determines a morphism X→Y.
Thus many map questions can be converted into ring questions. Dominance, closed embeddings and products have clean algebraic translations.
3. Closed embeddings correspond to surjective coordinate-ring maps
If Y⊂X is a closed subvariety, the inclusion i:Y→X pulls functions back by restriction.
The map i*:k[X]→k[Y] is surjective, because k[Y] is k[X] modulo the ideal of functions vanishing on Y.
Conversely, a surjective k-algebra map A→B between finitely generated reduced algebras corresponds contravariantly to a closed embedding Spec-style or variety-style in the classical affine setting.
Example: the y-axis Y=V(x)⊂A² has k[Y]=k[x,y]/(x)≈k[y]. The quotient map k[x,y]→k[y] mirrors the geometric inclusion.
4. Dominant morphisms correspond to injective pullbacks for irreducible varieties
A morphism φ:X→Y of irreducible varieties is dominant when its image is Zariski-dense in Y.
If φ is dominant, then φ*:k[Y]→k[X] is injective: a regular function that pulls back to zero vanishes on a dense image and hence vanishes on all of Y.
Conversely, in the affine irreducible setting, injectivity of the coordinate-ring pullback implies dominance.
Dominance is therefore a global geometric statement encoded by the absence of a nonzero algebraic relation killed by pullback.
5. Rational maps are morphisms defined on dense open subsets
A rational map X⇢Y is an equivalence class of morphisms defined on nonempty dense open subsets of an irreducible variety X that agree wherever both are defined.
For affine space, rational functions are quotients f/g in the function field k(X), with g nonzero.
The map A¹⇢A¹ given by x↦1/x is regular on D(x)=A¹\{0} but not at x=0.
A rational map can be birational: it can have a rational inverse even if neither direction is globally regular everywhere. Birational geometry studies varieties up to this dense-open equivalence.
The distinction matters: a formula that works generically may still have an indeterminacy locus that controls important geometry.
6. Projective morphisms are checked chart by chart
A morphism into projective space can often be written
p↦[F₀(p):⋯:F_m(p)]
where the Fᵢ are homogeneous of the same degree in projective coordinates and never vanish simultaneously on the domain.
On the chart F_j≠0, divide by F_j and obtain ordinary regular affine coordinate functions.
If all Fᵢ vanish at a point, the expression is not a morphism there. It defines at best a rational map with a base or indeterminacy locus.
This chartwise test keeps projective notation honest: [0:⋯:0] is never a permitted output.
7. Local rings keep only information visible near one point
Let X be affine and p∈X. Let m_p⊂k[X] be the maximal ideal of functions vanishing at p.
The local ring at p is
O_{X,p}=k[X]_{m_p}.
Localising at m_p allows every function not vanishing at p to become invertible. Such a denominator remains nonzero on some Zariski neighbourhood of p.
The local ring therefore forgets global behaviour irrelevant to p while retaining germs of regular functions near p.
Its unique maximal ideal m_pO_{X,p} consists of germs vanishing at p.
8. Dimension can be measured locally
The local dimension dim_pX is the Krull dimension of O_{X,p}. It records the maximum length of chains of irreducible subvarieties passing through p, expressed through chains of prime ideals in the local ring.
For an irreducible variety, the global dimension is constant as a top-dimensional invariant, but local dimension becomes especially useful on reducible algebraic sets where components of different dimensions may meet.
At a point lying on several components, local dimension is the maximum dimension of the components passing through that point.
Dimension is intrinsic. It does not depend on one chosen embedding into affine or projective space.
9. The Zariski tangent space comes from first-order functions
Let (O_{X,p},m) be the local ring. The quotient m/m² records functions vanishing to first order at p, modulo those vanishing to at least second order.
The Zariski cotangent space is m/m². The Zariski tangent space is its k-dual when p is a k-rational point:
T_pX=Hom_k(m/m²,k).
Its dimension is called the embedding dimension of the local ring.
This definition is intrinsic and makes sense even when X is not presented as a smooth-looking subset of Euclidean space.
10. The Jacobian computes the tangent space from equations
Suppose X=V(f₁,…,f_r)⊂Aⁿ and p∈X. Form the Jacobian matrix
J(p)=(∂fᵢ/∂xⱼ)(p).
Then the Zariski tangent space is the kernel of the linear map defined by J(p):
T_pX={v∈kⁿ : J(p)v=0}.
Therefore dim T_pX=n−rank J(p).
This formula is independent of the chosen generating set once all defining relations are interpreted correctly. Adding redundant equations cannot change the intrinsic tangent space.
MIT’s algebraic-geometry notes develop tangent spaces, differentials and smooth points through this algebraic/Jacobian interface: MIT 18.721 Algebraic Geometry notes.
11. Worked smooth example: the parabola has one-dimensional tangent spaces
Let X=V(f) with f(x,y)=y−x².
The gradient is ∇f=(−2x,1). It never vanishes.
At p=(a,a²), the tangent-space equation is
−2a·u+v=0.
This is a one-dimensional linear subspace of k². The variety itself has dimension 1, so tangent dimension and local dimension agree.
The tangent direction is generated by (1,2a), matching the derivative of the parametrisation t↦(t,t²).
12. Regular local rings define nonsingular points intrinsically
A Noetherian local ring (A,m) is regular when
dim A=dim_k(m/m²)
for the appropriate residue field formulation.
A point p of a classical variety is nonsingular or regular when O_{X,p} is a regular local ring.
Over an algebraically closed field of characteristic zero, the familiar Jacobian criterion detects this condition for varieties presented by polynomial equations.
The intrinsic definition is important because it survives changes of embedding and coordinate system.
13. A singular point has excess first-order directions
At a singular point, the tangent space is larger than the local dimension:
dim T_pX > dim_pX
in the classical variety setting.
Geometrically, the first-order equations fail to constrain the ambient directions as strongly as they do at a smooth point.
This can happen because two branches meet, because one branch develops a cusp, because a cone collapses many directions at a vertex, or because hidden nilpotent structure thickens the point.
“Singular” therefore describes local algebraic regularity, not one particular visual defect.
14. Worked singular example: the cusp y²=x³
Let C=V(f) with f=y²−x³.
The gradient is ∇f=(−3x²,2y). At the origin it is (0,0).
Therefore the Jacobian imposes no first-order linear equation at (0,0), so
T₀C=k².
But C is an irreducible curve of dimension 1. The tangent space has dimension 2, so the origin is singular.
A parametrisation is t↦(t²,t³). Its derivative at t=0 is (0,0), which explains why the parametrisation does not provide a nonzero first-order tangent vector there even though the algebraic tangent space is two-dimensional.
15. The tangent cone sees the first nonzero homogeneous terms
Translate a singular point to the origin and expand the defining equations into homogeneous pieces. The lowest-degree nonzero pieces define the tangent cone.
For the cusp y²−x³, the lowest-degree term is y², so the tangent cone is the doubled line y=0.
The ordinary Zariski tangent space is obtained from degree-one terms. Since the cusp has no nonzero linear term, that tangent space is the entire plane.
The tangent cone contains more refined information: it records that the singular branch approaches a single tangent direction with multiplicity.
16. A node has two tangent directions
Consider N=V(y²−x²−x³). At the origin the gradient vanishes, so the origin is singular.
The lowest-degree homogeneous part is y²−x²=(y−x)(y+x).
Thus the tangent cone is the union of the two lines y=x and y=−x.
Unlike the cusp, which has one doubled tangent direction, the node has two distinct branch directions.
Both are singular curves, but their local geometry is different. A useful singularity analysis therefore goes beyond the binary label singular/nonsingular.
17. Reducible crossings are singular too
For X=V(xy)⊂A², the components are the two coordinate axes.
At a point (a,0) with a≠0, the gradient of xy is (0,a), so the tangent space is one-dimensional and the point lies smoothly on the x-axis component.
At the origin, the gradient is zero, giving tangent space k² while the local dimension is 1.
The origin is singular because two irreducible components cross there.
This example also shows that singularity is local: the same algebraic set can be smooth at most points and singular at a small closed subset.
18. The singular locus is itself algebraic
For a hypersurface X=V(f) in characteristic zero, singular points satisfy
f=∂f/∂x₁=⋯=∂f/∂xₙ=0.
Thus the singular locus is cut out by polynomial equations and is Zariski-closed.
For systems of equations, one imposes rank conditions on the Jacobian. Rank≤r conditions are expressed by vanishing of minors of appropriate size.
So the place where smoothness fails is itself amenable to algebraic computation.
In positive characteristic, derivative tests can behave differently—for example pth powers have zero derivative—so the field and scheme-theoretic smoothness hypotheses must be kept explicit.
19. Worked projective singularity check
Let C⊂P² be the projective cubic
F(X,Y,Z)=Y²Z−X³=0.
Its partial derivatives are F_X=−3X², F_Y=2YZ and F_Z=Y².
At [0:0:1], all three partial derivatives vanish, so this point is singular.
On the affine chart Z=1 this is exactly the cusp y²=x³.
For a projective hypersurface, checking all homogeneous partial derivatives provides a chart-independent way to locate singular points under the standard characteristic hypotheses.
20. Smoothness is preserved by isomorphism
If X and Y are isomorphic varieties and p∈X maps to q∈Y, their local rings are isomorphic:
O_{X,p}≈O_{Y,q}.
Therefore regularity, local dimension and embedding dimension are preserved.
A singularity cannot be removed by merely changing coordinates through an algebraic isomorphism.
It can sometimes be resolved or normalised by replacing the variety with a different variety mapping to it. That operation is not an isomorphism at the singular point.
21. Dimension has several equivalent faces for affine varieties
For an irreducible affine variety X over k, the following dimensions agree:
- the geometric dimension from chains of irreducible closed subsets;
- the Krull dimension of k[X];
- the transcendence degree trdeg_k k(X) of the function field.
For Aⁿ, k(Aⁿ)=k(x₁,…,xₙ), whose transcendence degree is n.
For the parabola, k(X)≈k(t), so dimension is 1. For the surface z=xy in A³, the coordinate ring is k[x,y,z]/(z−xy)≈k[x,y], so dimension is 2.
These equivalences let one choose the dimension language best suited to the problem.
22. Codimension is controlled by height
If Y⊂X is an irreducible closed subvariety, its codimension in X is dimX−dimY.
Algebraically, the corresponding prime ideal has a height measuring the maximum length of prime chains beneath it.
In a polynomial ring k[x₁,…,xₙ], the principal ideal generated by a nonzero nonunit irreducible polynomial has height 1. An irreducible hypersurface therefore has codimension 1.
Several equations may define codimension less than their number when the equations are dependent or share components.
“Number of equations” becomes trustworthy only after the relevant independence or height conditions have been checked.
23. Jacobian rank explains the expected local dimension
For X⊂Aⁿ cut out locally by equations f₁,…,f_r, the tangent dimension is n−rankJ(p).
If p is smooth and the local codimension is c, then rankJ(p)=c and
dim_pX=n−c.
This is the algebraic analogue of the regular-level-set principle from differential geometry.
The analogy has limits. Algebraic singularities, field characteristic and nonreduced structure require commutative algebra beyond ordinary real-calculus rank tests.
Still, the shared pattern is valuable: independent local constraints remove independent tangent directions.
24. Fibres reveal how dimension is distributed by a map
For a morphism φ:X→Y and q∈Y, the fibre is
X_q=φ⁻¹(q).
Example: projection π:A²→A¹, (x,y)↦x. Every fibre π⁻¹(a) is a copy of A¹, so dimX=2, dimY=1 and generic fibre dimension=1.
For a dominant morphism of irreducible varieties, the generic fibre has dimension dimX−dimY under the standard finite-type hypotheses.
Special fibres can be larger. Fibre dimension is upper semicontinuous in suitable algebraic families: dimension jumps occur on special closed loci rather than randomly.
This is one route from static varieties to families and moduli problems.
25. Worked fibre-dimension example with a jump
Let X=V(tx)⊂A² with coordinates (t,x), and project to the t-axis.
If t=a≠0, the equation ax=0 forces x=0, so the fibre is one point, dimension 0.
If t=0, the equation 0·x=0 imposes no condition, so the fibre is all A¹, dimension 1.
The special fibre at t=0 is larger than the generic fibres.
This example is reducible globally, but it cleanly illustrates why “dimension of fibre” is not always constant in a family.
26. Differentials are dual to maps on cotangent spaces
A morphism φ:X→Y with φ(p)=q induces a local-ring homomorphism O_{Y,q}→O_{X,p}.
It sends the maximal ideal m_q into m_p and therefore induces a linear map
m_q/m_q²→m_p/m_p².
Dualising gives the tangent map
dφ_p:T_pX→T_qY.
In affine coordinates, this is represented by the usual Jacobian matrix of φ’s coordinate functions.
Thus the familiar derivative is recovered from the functorial behaviour of local rings.
27. Bijective morphisms can fail to be isomorphisms at singularities
Consider ν:A¹→C where C=V(y²−x³),
ν(t)=(t²,t³).
This map is bijective on k-points over an algebraically closed field, but the inverse is not regular at the cusp.
Away from the origin, t=y/x. At the origin both numerator and denominator vanish, and no regular function on the cusp extends this inverse.
Algebraically, k[C]=k[t²,t³] is a proper subring of k[t]. The inclusion k[t²,t³]⊂k[t] corresponds to ν.
The affine line is the normalisation of the cusp. Normalisation repairs a particular integrality defect without being an isomorphism at the singular point.
28. Normal does not mean smooth
A domain is integrally closed if every element of its fraction field integral over the ring already lies in the ring. An irreducible variety is normal when its local rings are integrally closed domains.
Every smooth variety is normal under standard finite-type hypotheses, but the converse fails in dimension at least two: normal varieties can have singularities.
Normality rules out certain codimension-one and branch-gluing pathologies; it does not guarantee that every local ring is regular.
The cusp is not normal, and its normalisation separates the missing integral function t.
This prepares the next R24 cell, where local rings and spectra replace classical varieties as the basic language.
29. Resolution replaces a singular variety by a smoother model
A resolution of singularities is, roughly, a proper birational morphism from a smooth variety to a singular one that is an isomorphism over a dense smooth region.
In characteristic zero, Hironaka’s theorem guarantees resolution for algebraic varieties under broad finite-type hypotheses. Positive characteristic remains substantially more difficult in higher dimensions.
Blow-ups are a central local operation: they replace a subvariety by the projectivised directions normal to it and can separate branches or simplify singularity structure.
Resolution is not “changing coordinates until the singularity disappears”. It changes the variety through a birational morphism.
For a deeper boundary marker, see MIT material on resolution and singularities: Resolution of Singularities notes.
30. A dependable singularity-and-dimension workflow
First state the base field, ambient space and defining ideal. Remove no equations merely because they appear redundant until their ideal consequences are understood.
At a point p, compute the local dimension or know the dimension of the component containing p. Form the Jacobian and find T_pX.
If tangent dimension equals local dimension under the standard classical hypotheses, the point passes the regularity test. Excess tangent dimension signals singularity.
For a singular point, inspect the lowest-degree homogeneous terms to find the tangent cone. Factor those terms to distinguish multiple tangent directions from repeated ones.
For maps, distinguish morphism, rational map, dominant map, closed embedding and birational map. Then inspect fibres and tangent maps rather than inferring behaviour from a formula alone.
Finally, keep regularity, normality and smoothness as distinct properties. Each answers a different local-algebraic question.
31. Common misconceptions
A singularity is a point where the graph has a corner. No. Singularity is defined through local algebra or smoothness, and many algebraic singularities are not described by ordinary graph corners.
Gradient zero always means singular without hypotheses. For a reduced hypersurface in the standard characteristic-zero setting, yes; in positive characteristic or nonreduced settings one must use the correct scheme-theoretic criterion.
Dimension equals variables minus equations. Only when the equations impose the expected independent conditions.
Bijective morphism equals isomorphism. The cusp normalisation is a standard counterexample.
Normal equals smooth. Smooth implies normal in the standard variety setting, but normal singular varieties exist.
32. Independent practice: twenty questions
- For X=V(y−x²), compute T_(a,a²)X.
- Why is every point of the parabola nonsingular?
- For C=V(y²−x³), compute the tangent space at the origin.
- Find the tangent cone of y²−x³ at the origin.
- For N=V(y²−x²−x³), find its tangent-cone directions at the origin.
- Locate the singular point of V(xy)⊂A².
- For the sphere-like affine surface x²+y²+z²−1=0 over C, show no point is singular in characteristic zero.
- For the cone z²−x²−y²=0, test the origin for singularity.
- For F=Y²Z−X³ in P², verify [0:0:1] is singular.
- Explain why a closed embedding gives a surjective coordinate-ring homomorphism.
- What algebraic property of φ*:k[Y]→k[X] corresponds to dominance for affine irreducible varieties?
- Why is x↦1/x only a rational map A¹⇢A¹?
- Compute dim k[x,y,z]/(z−xy).
- Give a prime chain showing dim k[x,y]≥2.
- For π:A²→A¹, (x,y)↦x, what is each fibre dimension?
- For X=V(tx)→A¹_t, compare the fibre over t=1 with the fibre over t=0.
- Why is t↦(t²,t³) not an isomorphism onto the cusp?
- What ring inclusion represents the cusp normalisation?
- State the intrinsic regular-local-ring criterion for a nonsingular point.
- Explain why resolution of singularities is not merely a coordinate change.
33. Worked solutions and checks
1. For f=y−x², J=(-2a,1). The tangent vectors (u,v) satisfy −2au+v=0, so T_pX=span{(1,2a)}.
2. The gradient (−2x,1) never vanishes. Hence the tangent dimension is always 1, equal to the curve dimension.
3. The gradient of y²−x³ is (−3x²,2y), which vanishes at the origin. Thus J=0 and T₀C=k².
4. The lowest-degree nonzero homogeneous term is y², so the tangent cone is the doubled line y=0.
5. The quadratic part is y²−x²=(y−x)(y+x). The two tangent directions are y=x and y=−x.
6. For f=xy, ∇f=(y,x), which vanishes only at (0,0). Away from the origin one partial derivative is nonzero.
7. The gradient is (2x,2y,2z). A singular point on the surface would require x=y=z=0, but that point does not satisfy x²+y²+z²=1. Hence none exists.
8. For f=z²−x²−y², the gradient is (−2x,−2y,2z), which vanishes at the origin. The cone has dimension 2 but tangent space dimension 3 there, so the vertex is singular.
9. F_X=−3X², F_Y=2YZ, F_Z=Y². At [0:0:1] all vanish.
10. Functions on the closed subvariety are restrictions of functions on the ambient affine variety, so k[Y]=k[X]/I_X(Y), making the restriction map surjective.
11. Injectivity. A nonzero target function cannot pull back to zero on a dense image.
12. The denominator vanishes at x=0. The formula is regular only on D(x).
13. The quotient is isomorphic to k[x,y] by eliminating z, so its Krull dimension is 2.
14. (0)⊊(x)⊊(x,y) is a strict chain of prime ideals of length 2.
15. π⁻¹(a)={a}×A¹, so every fibre has dimension 1.
16. At t=1, x=0, so the fibre is one point and has dimension 0. At t=0 the equation disappears, so the fibre is A¹ and has dimension 1.
17. It is bijective on points but the inverse is not regular at the cusp. Algebraically the coordinate ring k[t²,t³] is not isomorphic to k[t] through the proposed inverse.
18. k[t²,t³]⊂k[t]. The larger ring is the integral closure of the cusp ring in its fraction field.
19. O_{X,p} is regular when its Krull dimension equals the dimension of its cotangent space m/m² over the residue field.
20. A coordinate change gives an isomorphism of the same local algebra and cannot change regularity. A resolution replaces X by another variety through a proper birational morphism.
34. Where the first R24 batch ends
The first four R24 cells now form one route: polynomial zero sets → ideals and coordinate rings → projective closure → morphisms, tangent spaces, singularities and dimension.
The next batch changes the level of abstraction. R24.05 introduces schemes and local algebra; R24.06 adds sheaves and cohomology; R24.07 turns toward rational and integral points; R24.08 develops computational and enumerative methods.
Sources and further study
See MIT 18.721 Algebraic Geometry notes for affine morphisms, local algebra, tangent spaces and dimension; MIT 18.721 Projective Algebraic Geometry for projective morphisms and global geometry; and MIT Resolution of Singularities notes for the deeper singularity-resolution boundary.