Hassett 2: Weak approximation for function fields

Weak Approxmation

We start with the diagram:

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

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

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

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

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

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 $s=s_j \pmod{\mathfrak{m}_{b_j}^{n_j+1}}$.

Remarks.

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.

Effectivity

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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