Math 254B. Homework #3

Due Friday 19 February, online

1. Show that Lemma 2 (from class on February 8 and 10) is false without the subscheme X. In other words, with notation as in the lemma, show that h_{k}(ϕ(P)) − h_{k}(P) is unbounded as P varies over (ℙ_{k}^{n} \ V)(\overline{k}) .

2. Let k be a number field. Let ϕ: ℙ_{k}^{n} −−→ ℙ_{k}^{m} be a rational map given by

ϕ = [ϕ_{0} : ⋯: ϕ_{m}] ,

where ϕ_{i} ∈ k[x_{0}, …, x_{n}] are homogeneous polynomials of degree d. Let A be the N-tuple consisting of all the coefficients of all of the ϕ_{i}, where we will consider A to be a point in ℙ^{N}(k) for some N , and define h_{k}(ϕ) = h_{k}(A) .

(a). Prove that there is a constant c, independent of P and ϕ, such that

h_{k}(ϕ(P)) ≤ d h_{k}(P) + h_{k}(ϕ) + c
for all P ∈ ℙ_{k}^{n}(\overline{k}) in the domain of ϕ.

(b). Find a (reasonably sharp) explicit expression for c, in terms of the given mathematical objects (but independent of P and ϕ).

3(nc). Let F be a field and let k = F (t) be the function field of ℙ_{F}^{1} . Show that, given a constant C, there is a quasi-projective scheme X_{C} over F with the property that there is an algebraic bijection

X_{C}(F) → {x ∈ k : H_{k}(x) ≤ C}.

State what you mean by “algebraic.” This should be done directly; i.e., without reference to Chow schemes or Hilbert schemes.

This problem may be graded somewhat subjectively, based on the quality of the answer and not just whether it meets the requirements.