INTRO TO ALGEBRAIC GEOMETRY, PROBLEM SET 7

Due Thursday November 4 in class (no lates). Hand in six of the following problems. Make sure you know how to do the non-scheme problems you skip. You’re strongly encouraged to collaborate (although write up solutions separately), and you’re also strongly encouraged to ask me questions (if you’re stuck, or if the question is vaguely worded, or if you want to try out an argument).

1. (a) Hartshorne Ex. I.5.1. Assume the characteristic is not 2. Locate the singular points. Which is which in Figure 1 below? (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}.

(b) Hartshorne Ex. I.5.2. Assume the characteristic is not 2. Locate the singular points in 𝔸^{3}. Which is which in Figure 2 below? (I forgot to give the equations!)

(c) Find the dimension of the Zariski-tangent space to the surfaces in (b) at the origin.

2. Hartshorne Ex. I.5.3: Multiplicities. Let Y ⊂ 𝔸^{2} be the curve defined by the equation f(x, y) = 0. Let P = (a, b) be a point of 𝔸^{2}. Make a linear change of coordinates so that P becomes the point (0, 0). Then write f as a sum f = f_{0} + f_{1} + ⋯ + f_{d}, where f_{i} is a homogeneous polynomial of degree i in x and y. Define the multiplicity of P on Y, denoted μ_{P} (Y), to be the least r such that f_{r} ≠ 0. (Note that P ∈ Y if and only if μ_{P} (Y) > 0.) The linear factors of f_{r} are called the tangent directions at P.

(a) Show that μ_{P} (Y) = 1 if and only if P is a nonsingular point of Y.

(b) Find the multiplicity of each of the singular points in problem 1 (a) above.

3. Intersection multiplicity. If Y, Z ⊂ 𝔸^{2} are two distinct curves, given by equations f = 0, g = 0, and if P ∈ Y ∩ Z, define the intersection multiplicity (Y · Z)_{P} of Y and Z at P to be the length of the O_{𝔸^{2},P} -module O_{𝔸^{2},P} /(f, g), or equivalently its dimension as a \overline{k}-vector space. (It is also equivalent to replace O_{𝔸^{2},(a,b)} with the power series ring \overline{k}[[x′, y′]], where x′ = x − a and y′ = y − b; this makes calculations easier.)

(a) Calculate the intersection number of y^{2} = x^{5} and xy = 0 at the origin. (Don’t bother justifying all steps; just show how you computed the number.)

(b) If P ∈ Y, show that for almost all lines L through P (i.e. all but a finite number), (L · Y)_{P} = μ_{P} (Y).

4. 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} x_{i}(∂f/∂x_{i}); this algebraic trick is called Euler’s Lemma.)

5. Consider the subvariety V of 𝔸^{3} cut out by the equation z^{2} = xy. Explain why dim V = 2. Let R = \overline{k}[x, y, z]/(z^{2} − xy) be its coordinate ring. Find (with proof) a subvariety W of codimension 1 whose coordinate ring is not of the form R/(f) for any function f ∈ R. (Not precise. Instead: show I(W) is not principal in R. Show that W can be expressed as V (f) for some f ∈ R. (f will cut out W with multiplicity 2.)

Date: October 28, 1999.

6. Suppose f : X → Y is a morphism of varieties, with f(p) = q. Show that there are natural morphisms T (f)^{∗} : 𝔪_{q}/𝔪_{q}^{2} → 𝔪_{p}/𝔪_{p}^{2} (the induced map on cotangent spaces) and T (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.

7. Nonsingular schemes (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] → Spec 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.

I should note here that the map on tangent spaces doesn’t always exist. Note interpretation: at 0, codim 1 point.

8. Fibred products of affine schemes. (This problem isn’t as long as it looks!) (a) Fibred products of sets. Suppose X, Y, and Z are sets, and f : X → Z and g : Y → Z are (set) maps. Define a fibred product of X and Y over Z X ×_{Z} Y as follows: it is a set W along with morphisms p_{1} : W → X and p_{2} : W → Y, such that

\begin{aligned}W & → & Y \\ ↓ & & ↓ \\ X & → & Z \end{aligned}

is a commutative diagram, and for any maps a : V → X, b : V → Y, such that f ∘ a = g ∘ b (i.e. the two maps V → Z are the same), there is a unique map c : V → W such that a = p_{1} ∘ c and b = p_{2} ∘ c, and vice versa. (Informal translation: “maps V → X, Y that agree on Z correspond precisely to maps V → W.”)

Prove that W′ = {(x, y) ∈ X × Y |f(x) = g(y) ∈ Z} is a fibred product of X and Y over Z (so the above definition is an abstract nonsense way of saying something simple). Prove that if W is any fibred product, then there is a natural bijection W = W′. (Note that if Z is a one-element set, then X ×_{Z} Y can be naturally identified with the product X × Y.)

(b) Schemes. Fibred products in arbitrary categories can be defined in the same way. Consider now the category of affine schemes (which is the same as the category of rings, with the arrows reversed). If X = Spec R, and Y = Spec S, Z = Spec T, and f and g are induced by f^{∗} : T → R and g^{∗} : S → R, explain why W = Spec R ⊗_{T} S is a fibred product of X and Y over Z. (You’ll have to describe the morphisms p_{1} and p_{2}.) Hence fibred products exist in the category of affine schemes. (One can then show that fibred products exist in the category of schemes using the same argument as our proof that products exist in the category of prevarieties; see Hartshorne pp. 87–88.)

(c) Varieties. In the category of varieties, suppose the fibred product W = X ×_{Z} Y exists. If |Q| denotes the underlying set of a variety Q, show that |W | = |X| ×_{|Z|} |Y |. (Translation: the points of the fibred product is/are the fibred product of the points.)

(d) Fibred products of varieties don’t always exist. An affine scheme Spec R corresponds to an affine variety over \overline{k} if R is a finitely generated \overline{k} algebra that is a domain. Find an example of schemes X, Y, Z that correspond to varieties, and morphisms between them, such that the fibred product X ×_{Z} Y in the category of affine schemes doesn’t correspond to a variety. Hint: look in positive characteristic; you can even take X, Y, and Z to be the affine line. (This can easily be extended to prove that fibred products don’t always exist in the category of (pre)varieties over \overline{k}.)

From David Sheppard: reducedness is not a problem. Better: let Z be the affine line, X be a double cover, and Y a point mapping to a point of Z that isn’t a branch point.

(e) A fun example, not for credit. Calculate Spec ℂ ⊗_{Spec ℝ} Spec ℂ.

(Note: diagrams from Hartshorne p. 36 must be pasted in.)