Hassett 2: Weak approximation for function fields

Weak Approxmation

We start with the diagram:

$\displaystyle \xymatrix{
X \ar[r] \ar[d] & \mathcal{X}\ar[d] \\
{\mathrm{Spec}}F \ar[r] & {\mathrm{Spec}}\mathcal{O}_F

where $ X(F)=\mathcal{X}(\mathcal{O}_F)$, $ X$ smooth projective over a number field, and $ \mathcal{X}$ an integral model of $ X$.

Definition. The $ F$-rational points of $ X$ satisfy weak approximation if for each $ \{v_j\}$ a finite set of places, with completions $ F_{v_j}$, and open sets $ U_j \subset X(F_{v_j})$, there exists an $ x \in X(F)$ with $ x \in U_j$ for each $ j$.

Note that for nonarchimedean places, $ \mathfrak{p}_j \in {\mathrm{Spec}}\mathcal{O}_F$, $ \mathbb{F}_{\mathfrak{p}_j}=\mathcal{O}_F/\mathfrak{p}_j$, then we have reduction maps

$\displaystyle X(F_{v_j})=X(\widehat{\mathcal{O}_{F,v_j}}) \xrightarrow{\rho_{j,n}} X(\mathcal{O}_F/\mathfrak{p}_j^{n+1}) \ni s_j. $

The basic open subsets $ \rho_{j,n}^{-1}(s_j)$ have `fixed reduction modulo $ \mathfrak{p}_j^{n+1}$'.

Remarks. By Hensel's lemma, $ x_j \in X(\mathbb{F}_{\mathfrak{p}_j})$ gives a point in $ X(F_{v_j})$ if $ x_j$ is smooth.

If $ \mathcal{X}$ is regular, then if $ x_j \in X(\mathbb{F}_{\mathfrak{p}_j})$ comes from a point in $ X(F_{v_j})$, then $ x_j$ is regular.

Function field analog

Now consider the diagram

$\displaystyle \xymatrix{
X \ar[r] \ar[d] & \mathcal{X}\ar[d] \\
{\mathrm{Spec}}F \ar[r] & B

where $ B$ is a smooth projective curve over $ \mathbb{C}$, $ F=\mathbb{C}(B)$, and $ X$ a smooth projective variety over $ F$ with a regular projective model $ \mathcal{X}\to B$. Fix a finite set $ \{b_j\} \subset B$, $ x_j \in \mathcal{X}_{b_j}$ smooth points, and local Taylor series data at these points, $ s_j \in \mathcal{X}(\mathcal{O}_{B,b_j}/\mathfrak{m}_{b_j}^{n_j+1})$, $ s_j(b_j)=x_j$.

Definition. $ \mathcal{X}$ satisfies weak approximation if for any such set of data there exists $ s:B \to X$ so that $ s=s_j \pmod{\mathfrak{m}_{b_j}^{n_j+1}}$.


  1. $ \mathcal{X}$ satisfies weak approximation if and only if for each regular model $ \mathcal{X}_1 \to \mathcal{X}$, and points $ \{b_j\} \subset B$ and smooth points $ x_j' \in (\mathcal{X}_1)_{b_j}$, there exists a section $ s:B \to \mathcal{X}_1$ with $ s(b_j)=x_j'$.
  2. If $ \mathcal{X}_1,\mathcal{X}_2$ are models of $ X$, then $ \mathcal{X}_1$ satisfies weak approximation if and only if $ \mathcal{X}_2$ does, so it makes sense to say $ X$ satisfies weak approximation.
  3. $ F$-rational varieties satisfies weak approximation.

Rationally connected case

Let $ X \to F=\mathbb{C}(B)$ be rationally connected, with model $ \mathcal{X}\to B$. Here we have the theorem:

Theorem. [Graber, Harris, Starr; Kollár] There exists a section $ s:B \to \mathcal{X}$. Choose points $ \{b_j\} \subset B$ such that the fibres $ \mathcal{X}_{b_j}$ are smooth, and choose points $ x_j \in \mathcal{X}_{b_j}$; then there exists a section $ s:B \to X$ with $ s(b_j)=x_j$.

This will not give Taylor series data, because once one blows up to get the second-order Taylor series, the fibres are no longer irreducible.

All the fibers of $ \mathcal{X}\to B$ are rationally chain connected, except for the degenerate fibers (e.g., reducible fibers), which might have to go through singular points. Also, for example, the cone over an elliptic curve $ x^3+y^3+z^3=0$ is rationally chain connected but is not itself rationally connected.

Problem. Let $ X/F$ be a smooth projective variety, $ F=\mathbb{C}(B)$, $ B$ a curve. If $ X$ is rationally connected, show that $ X$ satisfies weak approximation.


Problem. Given $ b_j \in B$, $ D_{jk} \in \mathcal{X}_{b_j}$ of multiplicity one, does there exist an effective curve class $ [M]$ such that $ [M]\cdot \mathcal{X}_b=m$, and $ [M]\cdot D_{ij}=m$.

Let $ Y$ be a projective smooth variety over $ \mathbb{C}$. We have $ NS(Y) \subset H^2(Y,\mathbb{Z})$, the Néron-Severi group, and $ N_1(Y) \subset H_2(Y,\mathbb{Z})$, the $ 1$-cycles. We have the cone $ \overline{NE}^1(Y) \subset NS(Y)_\mathbb{R}$, the cone of effective divisors; we also have the cone of moving curves $ \overline{{\mathrm{Mov}}}_1(Y) \subset N_1(Y)_\mathbb{R}$, consisting of cycle classes $ [M]$ such that $ M$ is irreducible and passes through the generic point of $ Y$.

Given an effective divisor $ D$ and a moving class $ M$, then $ D \cdot M \geq 0$.

Note that $ \overline{NE}^1(Y) \subset \overline{{\mathrm{Mov}}}_1(Y)^*$, the dual cone.

Theorem. [Demailly, Peternell] Equality holds, $ \overline{NE}^1(Y)=\overline{{\mathrm{Mov}}}_1(Y)^*$.

As an application, this allows us to find $ [M] \in \overline{{\mathrm{Mov}}}^1(\mathcal{X})$ with the desired intersection properties.

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