Graber: Rationally Connected Varieties


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

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

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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