Math 254B. Homework #9

Due Thursday, 13 May, online

1(nc). Let k be a number field, let Y = Spec 𝒪_{k}, let V be a projective variety over k, let π_{1} : X_{1} → Y and π_{2} : X_{2} → Y be projective models for V , and let ℒ_{1} and ℒ_{2} be metrized line sheaves on models X_{1} and X_{2} , respectively, that agree on V . Show that

h_{ℒ_{1}} (P) = h_{ℒ_{2}} (P) + O(1) for all P ∈ V (k̄) .

Do this without using the “height machine” or results on heights proved using Weil functions. (So, you should use only the definition of h_{ℒ_{i}} from class on 26 April, and algebraic geometry on regular arithmetic surfaces.)

2. Let X be a nonsingular quasi-projective variety over an algebraically closed field, let W be a nonsingular subvariety of X, and let Y and Z be cycles on X. The intersection product W . Y is an element of CH(X) , but it may also be regarded as an element of CH(W) since all of its terms are subvarieties of W . To make things clearer, let (W.Y)_{W} denote the latter possibility. So, one can intersect Y and Z on X, and then intersect with W and regard it as an element of CH(W) : (W . (Y . Z))_{W} , or one can intersect W with Y and (separately) with Z, regard W . Y and W . Z as elements of CH(W) , and take their intersection product in CH(W) : (W . Y)_{W} . (W . Z)_{W} .

Show that (W . (Y . Z))_{W} = (W . Y)_{W} . (W . Z)_{W} (as elements of CH(W) ). You may assume that CH(W) → CH(X) is injective (shifting degrees by a con-

stant).

3. Let k be a number field. Find a bound B, depending only on k, such that every metrized line sheaf ℒ on Spec 𝒪_{k} with deg ℒ ≥ B has a nonzero global section s such that ∥s∥_{v} ≤ 1 for all v ∈ M_{k}^{∞}. (Such sections are called small sections of ℒ .) [Hint: Minkowski.]

4. Recall that the following was stated without proof on 5 May:

Proposition. Let X be a regular arithmetic surface, let D be an arithmetic divisor on X, and let f ∈ K(X)^{∗}. If D and (f) intersect properly, then (D. (f)) = 0 .

For a proof, it was asserted that it suffices to show this (a) when D = (Z, 0) , where Z is an irreducible component of a fiber of X → Spec ℤ, and (b) when D = (D_{fin}, g_{D}) , where D_{fin} is the prime divisor corresponding to a point P ∈ X(\overline{ℚ}) and \frac{1}{2}g_{D} is any smooth Weil function for P .

Give details on this reduction. (You do not need to prove (a) or (b), though.) Note that the intersection pairing (D. E) is bilinear when defined.

5(nc). Let 𝒪(1) denote the metrized line sheaf on ℙ_{ℤ}^{1} with Fubini–Study metric. Compute the arithmetic intersection number (𝒪(1)^{2}). [Hint: By the proposition mentioned in the previous problem, letting (𝒪(D) . 𝒪(E)) = (D . E) when D and E intersect properly gives a well-defined bilinear pairing P̂ic(X) × P̂ic(X) → ℝ. So it suffices to compute (D. E) for suitable arithmetic divisors D and E.]