Math 254B. Homework #8

Due Thursday 29 April, online

1. Let k be a number field or a function field in one variable over a field F , and let Y be as defined in class for k. Let V be a variety over k, and let X_{1} and X_{2} be models for V over Y . Let ℒ_{1} and ℒ_{2} be line sheaves on X_{1} and X_{2} , respectively, which agree on V . Show that ℒ_{1} and ℒ_{2} agree at almost all places of k; i.e., there is a nonempty open subscheme Y′ of Y such that

X_{1} ×_{Y} Y′ ≅ X_{2} ×_{Y} Y′

(compatible with the isomorphisms of their respective generic fibers with V ), and such that the restrictions of ℒ_{1} and ℒ_{2} correspond under this isomorphism.

2. Let k be a number field, a function field, or the completion of one of these fields at one of their given places. Let f: V → W be a morphism of complete varieties over k. Let D be a Cartier divisor on V whose restriction to the generic fiber of f is effective, and let λ_{D} be a Weil function for D. Show that there is a proper birational morphism g: W′ → W , a Cartier divisor E on W′, and a Weil function λ_{E} for E, with the following property. For all P ∈ (V \ Supp D)(M) and Q ∈ (W′ \ Supp E)(M) satisfying f(P) = g(Q) , we have λ_{D}(P) ≥ −λ_{E}(Q) .

If you aren’t familiar with blowings-up, you may assume that W is nonsingular, in which case the morphism g is not necessary. Or, at your option, you may use the following fact. Let E be a proper closed subset of W . Then there is a proper birational morphism g: W′ → W and an effective Cartier divisor E′ on W′ such that g(Supp E′) = E (as closed subsets of W ). (Here g is the blowing-up of W along E.)

3. Let k be a number field, let Y = Spec 𝒪_{k}, let n be a positive integer, and let h_{𝒪(1),FS} denote the height defined relative to the metrized line sheaf 𝒪(1) on ℙ_{Y}^{n} with Fubini– Study metrics at all archimedean places. Write an explicit expression for h_{𝒪(1),FS} in terms of homogeneous coordinates, and find explicit, best possible constants c and c′ (depending on n and k) such that

c ≤ h_{𝒪(1),FS} (P) − h_{k}(P) ≤ c′ for all P ∈ ℙ^{n}(k) .