Graber: Rationally Connected Varieties

Introduction

Definition. A field $K$ is quasi-algebraically closed ($C_1$) if any polynomial $F \in K[x_0,\dots,x_n]$ with $\deg F \leq n$ has a root in $K$.

Any finite field is quasi-algebraically closed, as is any function field $\overline{k}(C)$ of a curve over an algebrically closed field. This implies that any Laurent series ring $\overline{k}[[t]]$ is $C_1$.

These generalize to the following three possible definitions:

Definition. A projective variety $X$ is rationally connected (RC) if any two general points $p,q \in X$ can be joined by a rational curve $f:\mathbb{P}^1 \to X$.

Definition. $X$ is rationally chain connected (RCC) if any two general points $p,q \in X$ can be joined by a chain of rational curves.

Definition. $X$ is separably rationally connected (SRC) if $X$ is normal and there exists a rational curve $f:\mathbb{P}^1 \to X_{\text{sm}}$ such that $f^*TX$ is ample, i.e. $f^* TX=\bigoplus \mathscr{O}(d_i)$, $d_i>0$.

Theorem. If $X$ is smooth projective over $\mathbb{C}$, then $X$ is rationally connected (RC) if and only if $X$ is rationally chain connected (RCC) if and only if $X$ is separably rationally connected (SRC).

In characteristic zero, $X$ a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ is rationally connected if and only if $d \leq n$.

The main result we will consider is the following:

Theorem. [G, Harris, Starr] If $k=\overline{k}$, ${\mathrm{char}}k=0$, then any rationally connected variety over $k(C)$ for $C$ a curve has a rational point.

Remark. In characteristic $p$, the same is true for SRC. (de Jong, Starr)

Theorem. Over $\overline{k}((t))$, any rationally connected variety has a rational point.

This follows from the function field case. Geometrically, you can find a nonsingular integral model over $k[[t]]$; to find a section, it is equivalent to find a reduced component of the central fiber.

Theorem. [Ernault] If $X/\mathbb{F}_q$ is smooth, projective, and geometrically rationally chain connected, then $X$ has a rational point.

Proof, a Beginning

We now prove the theorem that an RC variety over $k(C)$ has a rational point.

Choose an integral model $f:X \to B$. First, we reduce to the case $B\cong \mathbb{P}^1$. By restriction of scalars, have a map $X \to B \to \mathbb{P}^1$; the fibres of $X \to \mathbb{P}^1$ are products of fibres of $f$, and the product of rationally connected varieties is rationally connected.

Next, choose a curve $C \subset X$ which dominates $\mathbb{P}^1$. Deform $C$ and specialize until it breaks off a section. We have a map of moduli spaces

$$\{$$curves in $X$$$\} \xrightarrow{\pi_f} \{$$coverings of $\mathbb{P}^1$$$\}.$$

The latter is built out the data of the branch points plus monodromy; fixing the genus of the curve and the degree of the cover, we know that this moduli space is irreducible. It is possible to degenerate any branched cover of $\mathbb{P}^1$ to a reducible cover with a section as one component. It is enough to show that this map $\pi_f$ on moduli spaces is surjective. Now it is just a matter of tracing the monodromy as the branch points move around (at least in characteristic zero). In characteristic $p$, it can also be done (look, for example, at $\mathbb{P}^1 \to \mathbb{P}^1$).

To make all of this more precise, we look at stable maps. Given $X$ a smooth projective variety over $\mathbb{C}$, and given $g \in \mathbb{N}$ and $\beta \in H_2(X,\mathbb{Z})$, we construct a space $\overline{\mathcal{M}_g}(X,\beta)$ consisting of maps $f:C \to X$ such that $C$ is a connected nodal curve of arithmetic genus $g$, $f$ is a morphism, and $f_*(C)=\beta$, together with a stability criterion.

This space $\overline{\mathcal{M}_g}(X,\beta)$ is projective, and given any morphism $f:X \to Y$, one has an induced map $M_f:\overline{\mathcal{M}_g}(X,\beta) \to \overline{\mathcal{M}_g}(Y,f_*\beta)$ from $C \to X \to Y$. If $Y=\mathbb{P}^1$, then $\overline{\mathcal{M}_g}(\mathbb{P}^1,d)$ is a compactification of the space of branched covers. We want to show that $M_f$ is surjective. (We only need the coarse moduli space; in fact, at least in characteristic zero we have a Deligne-Mumford stack.)

When $K$ is $C_1$, the bound $d \leq n$ is sharp. If $K$ is not algebraically closed, then pick a finite extension $K \subset L$, and consider $N_L:L \to K$; this has a polynomial of degree $[L:K]$ with no nontrivial zeros; therefore it is impossible to get a larger class of hypersurfaces.

Is the notion of rational connectivity sharp? In the case of finite fields, we also get: if $X/\mathbb{F}_q$ smooth projective, and either $H^i(X,\mathscr{O}_X)=0$ for $i>0$ or $H^0(X,\bigwedge^i \Omega_X)=0$ for $i>0$, then $X$ has a rational point. (These two are equivalent in characteristic zero.) The analog over function fields is false. In particular, there exists a family $X$ of Enriques surfaces over $k(C)$ ($C$ is a curve over $\mathbb{C}$) which has no section.

The general statement: Let $\pi:X \to M$ be a proper morphism of varieties over $\mathbb{C}$. Suppose that for all maps $f:C \to M$, there exists a pullback $\widetilde{f}:C \to X$. It would suffice for there to exist $Z \subset X$ dominating $M$ such that the general fibre of $\pi\vert _Z$ is rationally connected. It is a theorem that this is necessary and sufficient (G, Harris, Mazur, Starr).

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