What is the dimension of a variety?
By now you can build affine varieties, read their coordinate rings, and pass to the function field of an irreducible one. The next question is the most basic of all: how big is V? A curve should be 1-dimensional, a surface 2-dimensional, but V lives inside the Zariski topology where open sets are enormous and the naive 'count the continuous parameters' intuition needs a precise algebraic home. The answer is the Krull dimension, and its beauty is that it is read off the ring alone, no calculus required.
The Krull dimension of V is the length d of the longest strictly increasing chain of irreducible closed subvarieties Z_0 ⊊ Z_1 ⊊ ... ⊊ Z_d ⊆ V. Picture a surface in 3-space: a point sits inside a curve which sits inside the surface, giving the chain point ⊊ curve ⊊ surface, length 2 — exactly the dimension we wanted. Dually, in the coordinate ring this is the longest chain of prime ideals, since each irreducible subvariety corresponds to a prime. So 'dimension' is a purely commutative-algebra invariant: the supremum of lengths of chains of primes in k[V].
Three other measurements quietly agree with this, and that agreement is a small miracle worth trusting. For an irreducible V over an algebraically closed field k, the Krull dimension equals the transcendence degree of the function field k(V) over k — that is, the number of algebraically independent rational functions you can choose. It also equals the dimension you would get from the Zariski tangent space at a generic point. For a curve, k(V) has transcendence degree 1: pick one coordinate function and every other rational function is algebraic over it. That these all coincide is what lets you compute dimension by whichever route is easiest.
The Zariski tangent space: calculus with no calculus
On a smooth manifold from Vol I you defined T_p M by curves or derivations and never worried whether the space had the right size. A variety can have corners and crossings, so we need a tangent notion that detects them. The trick is to define the tangent space using only the algebra of functions near p, mimicking the manifold definition through derivations. Let m be the maximal ideal of functions vanishing at p inside the local ring at p. Then m/m^2 is a finite-dimensional vector space over k — the cotangent space — and the Zariski tangent space T_p V is its dual (m/m^2)*.
Why m/m^2? An element of m is a function vanishing at p; squaring or multiplying two such functions vanishes 'to second order', so quotienting by m^2 throws away the higher-order behaviour and keeps exactly the linear part — the differential. This is the algebraic incarnation of 'df at p'. A linear functional on m/m^2 is precisely a derivation: a map satisfying the Leibniz rule, which is what a tangent vector is. So the definition that looked like sleight of hand recovers the honest tangent vectors you already know, but now it works at points where there is no manifold structure at all.
Smooth vs singular: the Jacobian criterion
A point p of V is smooth (or regular, or non-singular) when dim T_p V equals dim V at p, and singular otherwise — exactly the failure of the inequality above to be an equality. To test it by hand you do not need the abstract local ring: you need one matrix. If V is cut out in affine n-space by polynomials f_1, ..., f_r, form the Jacobian matrix of partial derivatives (∂f_i / ∂x_j) and evaluate it at p. The Jacobian criterion says p is smooth precisely when this matrix has rank n − d, the codimension of V.
This is the algebraic-geometry echo of the regular value theorem you met in Vol I differential topology: full-rank differential means the level set is locally a manifold of the expected dimension. The Jacobian rows are exactly the differentials df_i, and their span inside the cotangent space measures how many independent linear conditions actually cut at p. When the rank drops, fewer conditions bite, the tangent space is too big, and p is singular. The dictionary 'smooth point ↔ manifold point' is therefore not a vague analogy but a theorem with the same proof shape, run over the function ring.
Worked example: the nodal cubic y^2 = x^3 + x^2 in the affine plane.
Write it as f(x,y) = y^2 - x^3 - x^2 = 0. Variety has dimension d = 1, so n - d = 2 - 1 = 1.
Jacobian (one row, since r = 1): [ df/dx , df/dy ] = [ -3x^2 - 2x , 2y ].
The matrix has rank 1 (= n - d, SMOOTH) unless BOTH entries vanish:
2y = 0 => y = 0
-3x^2 - 2x = 0 => x = 0 or x = -2/3
Check which of these lie ON the curve f = 0:
(0, 0): 0 = 0 YES -> on the curve, Jacobian rank 0 -> SINGULAR (the node)
(-2/3, 0): 0 = (-2/3)^3 + (-2/3)^2 = 4/27 != 0 -> NOT on the curve, irrelevant
Conclusion: exactly one singular point, the origin, where the two branches cross.Blowing up: replacing a point by a line
Now the showpiece. We have located a singular point — what do we do with it? The blow-up is the central surgery for repairing singularities. Centred at the origin of the affine plane, the blow-up is the variety B = { ((x,y), [u:v]) : x v = y u } sitting inside (plane) × (projective line), together with the projection pi back down to the plane. The condition x v = y u says the point (x,y) lies on the line through the origin with direction [u:v]: away from the origin this line is forced, so pi is a bijection there.
Over the origin, though, every direction [u:v] satisfies the equation, so pi^{-1}(0) is an entire copy of the projective line — the exceptional divisor E. The blow-up is a birational map: it is an isomorphism away from the centre and replaces the single point 0 by the full set of tangent directions at 0. That is the key picture to carry: blowing up separates information that was crushed together at one point by spreading it out along E, one slot per direction. Crucially, B is itself smooth even though it maps onto the plane, and pi is dominant, so it does not change the function field — same field of rational functions, gentler geometry.
- Locate the singularity: run the Jacobian criterion to find the bad point and read off its tangent cone — for the node y^2 = x^3 + x^2 the lowest-degree part is y^2 - x^2 = (y-x)(y+x), so two distinct tangent lines cross at the origin.
- Blow up the centre: introduce the chart by setting y = t x (the slope coordinate on the exceptional line), so a point is recorded by its position x and the direction t it approached along.
- Substitute and divide out the exceptional divisor: plug y = t x into y^2 = x^3 + x^2 to get x^2 t^2 = x^3 + x^2, then cancel the x^2 that cuts out E, leaving the strict transform t^2 = x + 1.
- Check the result is smooth: the strict transform t^2 = x + 1 is a smooth curve (its Jacobian [-1, 2t] never vanishes), and it meets E = {x = 0} at the two points t = +1 and t = -1 — the two tangent directions, now pulled apart into two clean points.
Resolution, honest limits, and the road to Bézout
What just happened is a baby case of resolution of singularities: a single blow-up turned a singular curve into a smooth one mapping down to it. For curves over any field this always succeeds in finitely many blow-ups — you can untangle every node, cusp, and tacnode. In higher dimensions the celebrated theorem of Hironaka guarantees resolution by blow-ups too, but only in characteristic zero, and only as the conclusion of one of the longest, most demanding proofs in mathematics. In positive characteristic, resolution in dimension 3 and above is still partly open. Be honest about that: 'blow up until smooth' is a true and complete recipe for curves, a deep theorem for varieties over C, and a research frontier elsewhere.
Resolving singularities is also what makes intersection numbers behave, which is exactly where Guide 5 picks up. When two curves meet at a singular or tangential point, the naive count 'one point of intersection' lies — they may meet with high multiplicity. The intersection multiplicity is the precise local count, and blowing up is one clean way to see it: separate the branches, count the honest transverse intersections upstairs, and add the contribution of E. With dimension, smoothness, and the blow-up in hand, you are ready for the capstone — Bézout's theorem, which says two projective plane curves of degrees m and n meet in exactly m·n points once every multiplicity and point at infinity is counted correctly.