Math 256B. Homework 8

Due Wednesday 20 March

1(nc). Let ℒ be a line sheaf on a scheme X. Suppose that there exist global sections s ∈ Γ(X, ℒ) and t ∈ Γ(X, ℒ^{∨}) such that s ⊗ t maps to 1 under the canonical isomorphism ℒ ⊗ ℒ^{∨} →^{∼} 𝒪_{X} . Show that there exist isomorphisms ϕ: ℒ →^{∼} 𝒪_{X} and ψ: ℒ^{∨} →^{∼} 𝒪_{X} such that ϕ(s) = ψ(t) = 1 .

2. Let ℒ be an ample line sheaf on a projective variety X. Let V_{1}, …, V_{n} be closed subvarieties, with V_{i} ⊈ V_{j} for all i ≠ j. Then there exists a positive integer m and t_{1}, …, t_{n} ∈ Γ(X, ℒ^{⊗m}) such that t_{i}|_{V_{j}} ≠ 0 for all i ≠ j, and t_{i}|_{V_{i}} = 0 for all i.

Here if X is a scheme, if ℳ is a quasi-coherent sheaf on X, if Y is a closed subscheme of X with corresponding closed immersion i: Y → X, and if s is a global section of ℳ , then s|_{Y} is the global section of i^{∗}ℳ defined locally by s ⊗ 1 (via the isomorphism of (II, Prop. 5.2e)). Or, tensoring the natural map 𝒪_{X} → 𝒪_{Y} with ℳ gives a map ℳ → ℳ ⊗ 𝒪_{Y} , and s|_{Y} is the image of s under this map (note that ℳ ⊗𝒪_{Y} ≅ i^{∗}ℳ ). It is also true (and you may use without proof) that if Y is integral, and if η is its generic point, then s|_{Y} = 0 if and only if s_{η} ∈ 𝔪_{η}ℳ_{η}, where 𝔪 is the maximal ideal in the local ring 𝒪_{X,η}.

[Hint: Think of the prime avoidance lemma in commutative algebra.]

3. Let A_{0} be the subring of k[x, y] generated by the set of all homogeneous polynomials

of degree ≠ 1 , let A = (A_{0})_{x^{2}−1} , and let X = Spec A.

(a). Show that X is separated, noetherian, integral, and regular in codimension

one. (b). Show that X is not normal. (c). Show that the divisor (x−1) equals zero as a Weil divisor, but that x−1 is not a regular function on X. (Note that x−1 is an element of K(X) = k(x, y) .)