Math 254B. Homework #5

Due Thursday 18 March, online

1. Let E be the curve y^{2}z = x^{3} − xz^{2} in ℙ_{k}^{2} for some algebraically closed field k of characteristic ≠ 2 (as in Hartshorne II 6.10.2). Let Δ be the prime divisor consisting of the diagonal on E ×_{k} E (i.e., the image of the diagonal morphism (id, id) : E → E ×E, with reduced induced subscheme structure). Show that Δ (as a Weil divisor) is not linearly equivalent to a divisor of the form p^{∗}D_{1} + q^{∗}D_{2} . Here p, q: E ×_{k} E → E are the projection maps, and p^{∗}: Div(E) → Div(E ×_{k} E) and q^{∗}: Div(E) → Div(E ×_{k} E) are the maps taking a prime divisor P ∈ E to P × E and E × P , respectively. You may assume that E and E ×_{k} E are regular schemes.

2(nc). Let k be either a number field, a function field of transcendence degree 1 over a field F , or the completion of a field k_{0} of either of those two types at a place v ∈ M_{k_{0}} . Let X be a variety over k, and let v ∈ M_{k}. Show that, for any nonempty open affine U = Spec A in X and any isomorphism A ≅ k[x_{1}, …, x_{n}]/I (where I is an ideal in the polynomial ring k[x_{1}, …, x_{n}] ), the topology on U(ℂ_{v}) induced by the v-topology on X(ℂ_{v}) is the relative topology of U(ℂ_{v}) as a subset of ℂ_{v}^{n} = 𝔸^{n}(ℂ_{v}) .

3(nc). Let k be a field as in the previous problem, let U = Spec A be an affine variety over k, and let U_{i} = D(f_{i}) , i = 1, …, n, be an open cover of U by principal open affines. Show that a subset E ⊆ U(M) is affine M-bounded with respect to U if and only if it can be written as E = E_{1} ∪ ⋯ ∪ E_{n}, where E_{i} is affine M-bounded with respect to U_{i} for all i.