INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 15

RAVI VAKIL

Contents

1. Non-singularity 1 1.1. A more algebraic definition of nonsingularity; hence nonsingularity is

intrinsic 1

1.2. Examples 3 1.3. The singular points form a closed subset 4 2. Curves: Valuation rings and nonsingular points, take 1 4

Hand in problem sets at end.

Scheme aside: All varieties can be considered as schemes.

1. Non-singularity

Last day:

Definition. Let Y be a dimension d affine variety in 𝔸^{n} (with coordinates x_{1}, …, x_{n}. Suppose Y is defined by equations f_{1}, …, f_{t} (i.e. I(Y) is generated by the f_{i}; recall that any ideal in \overline{k}[x_{1}, …, x_{n}] is finitely generated!). Warning: We know that t is at least the codimension n−d, but they two aren’t necessarily equal! Then Y ⊂ 𝔸^{n} is nonsingular at a point p ∈ Y if the rank of the Jacobian matrix (∂f_{i}/∂x_{j}(p))_{i,j} is n − d.

Remark. Derivatives are just “formal”, i.e. dx^{n}/dx = nx^{n−1}, even in characteristic p; there are no limits here.

1.1. A more algebraic definition of nonsingularity; hence nonsingularity is intrinsic. Algebraic Definition. Let A be a noetherian local ring with maximal ideal 𝔪 and algebraically residue field \overline{k}. Then A is a regular local ring if dim\overline{k} 𝔪/𝔪^{2} = dim A. I should have made clear that the dimension here is just the dimension of 𝔪/𝔪^{2} as a \overline{k}-vector space.

The reason this definition will be relevant is:

Date: Thursday, October 26, 1999.

Theorem (*). Let Y ⊂ 𝔸^{n} be an affine variety. Let p ∈ Y be a point. Then Y is nonsingular at p if and only if the local ring O_{Y,p} is a regular local ring. Leave on board until proof is complete.

I promised you a proof, and I’ll delay the proof for a few minutes after I summarize last day some more.

Hence the concept of nonsingularity is intrinsic, so we can make the following definitions:

Definition. Let Y be any prevariety. Then Y is nonsingular at a point p ∈ Y if the local ring O_{Y,p} is a regular local ring; otherwise it is singular at p. Y is nonsingular if it is nonsingular at any point. Otherwise it is singular.

Remark. To check that something is singular, it is still easier to look at an affine cover and use the Jacobian definition.

Theorem (**). Let A be the localization of \overline{k}[x_{1}, …, x_{n}] at the origin, so A has dimension n. Then 𝔪/𝔪^{2} is naturally isomorphic to the vector space (α_{1}, …, α_{n}) ∈ \overline{k}^{n} (call it V), where points of the vector space can be associated with linear forms α_{1}x_{1} + … α_{n} + x_{n}. Hence A is a regular local ring.

Remark. 𝔪/𝔪^{2} is clearly basis-free.

Recap of proof of Theorem (**): The proof identified the vector space V of “hyperplanes through the origin” α_{1}x_{1} +⋯+α_{n}x_{n} = 0 with 𝔪/𝔪^{2}. The map from left to right is easy; the map from right to left is given by

f ∈ A ↦ (∂f/∂x_{1}(0, ..., 0), …, ∂f/∂x_{n}(0, ..., 0))

(where f vanishes at the origin). We checked that this is well-defined, i.e. if f ∈ 𝔪^{2}, then this is the 0-map; this was the chain rule.

Proof of Theorem (*). You’ll notice the similarity to the above proof.

Let p be a point (a_{1}, …, a_{n}) in 𝔸^{n}, and let 𝔞 = (x_{1} − a_{1}, …, x_{n} − a_{n}) be the maximal ideal corresponding to p in A = \overline{k}[x_{1}, …, x_{n}]. We define a linear map θ : A → \overline{k}^{n} by

θ(f) = (\frac{∂f}{∂x_{1}} (p), …, \frac{∂f}{∂x_{n}} (p))

for any f ∈ A. Notice that this is exactly the same map as in the proof about 𝔸^{n}. (You should really think of the \overline{k}^{n} as being the space of linear forms vanishing at the origin.) As we said before, θ induces an isomorphism θ′ : 𝔞/𝔞^{2} → \overline{k}^{n}.

Now let’s bring Y into the picture. Let 𝔟 be the ideal of Y in A, and let f_{1}, …, f_{t} be a set of generators of 𝔟. Then the rank of the Jacobian matrix J = (∂f_{i}/∂x_{j})(p)_{i,j} is just the dimension of θ(𝔟) as a subspace of \overline{k}^{n}. Using the isomorphism θ′, this is the same as the dimension of the subspace (𝔟 + 𝔞^{2})/(𝔞^{2}) of 𝔞/𝔞^{2}.

(You should check: R is a ring, 𝔪 is a maximal ideal, R_{𝔪} localization. Then R/𝔪 ≅ R_{𝔪}/𝔪R_{𝔪} ≅ \overline{k}, R/𝔪^{2} ≅ R_{𝔪}/𝔪^{2}R_{𝔪}, and 𝔪/𝔪^{2} ≅ 𝔪R_{𝔪}/𝔪^{2}R_{𝔪}. Exercise on Problem Set 8.)

Now the local ring O_{Y,p} of p on Y is obtained by modding out by 𝔟 and localizing at the maximal ideal 𝔞. Thus if 𝔪 is the maximal ideal of O_{Y,p}, then 𝔪/𝔪^{2} ≅ 𝔞/(𝔟 + 𝔞^{2}). Counting dimensions of vector spaces, dim 𝔪/𝔪^{2} + rkJ = n.

Now the dimension of the local ring O_{Y,p} (as a ring) is the dimension of Y (as a variety), so O_{Y,p} is regular if and only if dim 𝔪/𝔪^{2} = r. But this is equivalent to rkJ = n − r, which says that p is a nonsingular point of Y.

Important observation from Theorem (**). Notice that the elements of 𝔪/𝔪^{2} are naturally identified with linear functions on 𝔸^{n}. Now 𝔸^{n} can canonically be identified with the tangent space of 𝔸^{n} at the origin. So we’ve made an identification of 𝔪/𝔪^{2} with the cotangent space of 𝔸^{n} at the origin.

Based on this observation we made the following definition:

Definition. Let (A, 𝔪) be the local ring of a point p ∈ Y. Call 𝔪/𝔪^{2} the Zariski co-tangent space to Y at p, and (𝔪/𝔪^{2})^{∗} the Zariski tangent space.

Exercise. Suppose f : X → Y is a morphism of varieties, with f(p) = q. Show that there are natural morphisms f^{∗} : 𝔪_{q}/𝔪_{q}^{2} → 𝔪_{p}/𝔪_{p}^{2} (the induced map on cotangent spaces) and f^{∗} : (𝔪_{p}/𝔪_{p}^{2})^{∗} → (𝔪_{q}/𝔪_{q}^{2})^{∗} (the induced map on tangent spaces). (If you imagine what is happening on the level of tangent spaces and cotangent spaces of smooth manifolds, this is quite reasonable.) If ϕ is the vertical projection of the parabola x = y^{2} onto the x-axis, show that the induced map of tangent spaces at the origin is the zero map.

1.2. Examples. It isn’t hard to check for singular points, especially on hypersurfaces in 𝔸^{n}. For example, consider the plane curve y^{2} = x^{3} − x^{2}. (Do it.)

Exercise. Hartshorne Ex. I.5.1. Find the singular points, and sketch the following plane curves: (a) x^{2} = x^{4} + y^{4}, (b) xy = x^{6} + y^{6}, (c) x^{3} = y^{2} + x^{4} + y^{4}, (d) x^{2}y + xy^{2} = x^{4} + y^{4}. Hartshorne Ex. I.5.2, some two-dimensional examples.

Hypersurfaces in projective space are also easy, because there is a trick.

Exercise. Suppose the characteristic of \overline{k} is 0. Suppose a hypersurface Y ⊂ ℙ^{n} is given by f(x_{0}, …, x_{n}) = 0. Show that the locus of points p ∈ ℙ^{n} where ∂f/∂x_{i}(p) = 0 for all i are precisely the singular points of Y. (In particular, if ∂f/∂x_{i}(p) = 0 for all i, then f(p) = 0, i.e. p ∈ Y! To see why, calculate ∑_{i} ∂f/∂x_{i}.)

As an example, consider y^{2}z − x^{3} = 0. The only singular point is (x; y; z) = (0; 0; 1).

Scheme examples.

Exercise. (a) Show that both Spec ℤ and Spec ℤ[i] are nonsingular curves. (b) Let 𝔪 = (1 + i) in ℤ[i]. Then under the map f : Spec ℤ[i] → ℤ, f(𝔪) = (2). Check that the map on cotangent spaces (or equivalently, that the dual map on tangent spaces) is the zero-map. For all other primes of ℤ[i], calculate the map on cotangent spaces.

1.3. The singular points form a closed subset. Theorem. Let Y be a prevariety. Then the set of singular points Sing Y is a closed subset of Y.

In the proof, we’ll use the following (which will come up in the Commutative Algebra course).

Commutative Algebra Fact. If A is a noetherian local ring with maximal ideal 𝔪 and residue field \overline{k}, then dim\overline{k} 𝔪/𝔪^{2} ≥ dim A.

Remark. It is also true that Sing Y ≠ Y, but I won’t prove this fact.

Proof of Theorem. First step: reduce to the affine case. It suffices to show that for some open covering Y = ∪Y_{i} of Y, that Sing Y_{i} is closed for each i. So assume that Y is affine.

By the above C.A. fact, we know that the rank of the Jacobian matrix is at most n − r, so the set of singular points is the set of points where the rank is less than n − r. Thus Sing Y is the set defined by the ideal I(Y) together with all determinants of (n − r) × (n − r) submatrices of (∂f_{i}/∂x_{j})_{i,j}, which is a closed set.

2. Curves: Valuation rings and nonsingular points, take 1

Dimension 1 varieties, or curves, are particularly simple, and most of the rest of the course will concentrate on them.

We saw that nonsingularity has to do with local rings, so we’ll discuss onedimensional local rings.

First we’ll recall some facts about discrete valuation rings and Dedekind domains.

Definition. Let K be a field. A discrete valuation of K is a map v : K \ {0} → ℤ such that for all x, y non-zero in K, we have: v(xy) = v(x) + v(y), v(x + y) ≥ min(v(x), v(y)). Notice that the set R = {x ∈ K|v(x) ≥ 0} ∪ {0} is a subring of K; call this the discrete valuation ring, or DVR, of K. The subset 𝔪 = {x ∈ K|v(x) > 0} ∪ {0} is an ideal in R, and (R, 𝔪) is a local ring. A discrete valuation ring is an integral domain which is the discrete valuation ring of some valuation of its quotient field. If \overline{k} is a subfield of K such that v(x) = 0 for all x ∈ \overline{k} \ \{0\}, then we say v is a discrete valuation of K/\overline{k}, and R is a discrete valuation ring of K/\overline{k}.

I’ll need to patch this to prevent pathologies such as the “zero valuation”. More on this next day.

Example. Let K = \overline{k}(t), and for f ∈ K, let v(f) be the order of the zero of f at t = 0 (negative if f has a pole). Check all properties. Notice that discrete valuation ring of of v are those quotients of polynomials whose denominator doesn’t vanish at 0, i.e. \overline{k}[t]_{(t)}. In geometric language, it is the stalk of the structure sheaf of 𝔸^{1} at the origin.

Also, \overline{k}[t]_{(t)} is a discrete valuation ring: it is indeed an integral domain, and it is the valuation ring of some valuation in its quotient field \overline{k}(t).

Similarly, we could get other valuations by replacing 0 with any other element of \overline{k}. Have we found all the valuations? No:

Example. Let K = \overline{k}(t) as before. For f ∈ K, write f in terms of u = 1/t, and let v(f) be the order of zero of f at u = 0. Again, it is indeed a valuation, and it has geometric meaning. (Ask them.) It corresponds to the point of ℙ^{1} “at ∞” (when looking at it with respect to the t-coordinate).

Fact (that we’ll later prove). These are all the valuations of \overline{k}(t), the function field of ℙ^{1}. They naturally correspond to the points of ℙ^{1}.

Example. Let K = ℚ. (Ask for valuations.) If f ∈ ℚ, let v(x) be the highest power of 2 dividing x, so v(14) = 2, v(3) = 0, v(13/12) = −2. Check all properties. What’s the discrete valuation ring? Those fractions with no 2’s in the denominators. Geometrically, ℚ is the function field of Spec ℤ, and the valuations turn out to correspond to the maximal prime ideals of ℤ, i.e. the “closed points” of Spec ℤ.

Coming soon. Describing discrete valuation rings in many ways. Integral closure. Dedekind domains.