Definition. A field is quasi-algebraically closed () if any polynomial with has a root in .
Any finite field is quasi-algebraically closed, as is any function field of a curve over an algebrically closed field. This implies that any Laurent series ring is .
These generalize to the following three possible definitions:
Definition. A projective variety is rationally connected (RC) if any two general points can be joined by a rational curve .
Definition. is rationally chain connected (RCC) if any two general points can be joined by a chain of rational curves.
Definition. is separably rationally connected (SRC) if is normal and there exists a rational curve such that is ample, i.e. , .
Theorem. If is smooth projective over , then is rationally connected (RC) if and only if is rationally chain connected (RCC) if and only if is separably rationally connected (SRC).
In characteristic zero, a smooth hypersurface of degree in is rationally connected if and only if .
The main result we will consider is the following:
Theorem. [G, Harris, Starr] If , , then any rationally connected variety over for a curve has a rational point.
Remark. In characteristic , the same is true for SRC. (de Jong, Starr)
Theorem. Over , any rationally connected variety has a rational point.
This follows from the function field case. Geometrically, you can find a nonsingular integral model over ; to find a section, it is equivalent to find a reduced component of the central fiber.
Theorem. [Ernault] If is smooth, projective, and geometrically rationally chain connected, then has a rational point.
Proof, a Beginning
We now prove the theorem that an RC variety over has a rational point.
Choose an integral model . First, we reduce to the case . By restriction of scalars, have a map ; the fibres of are products of fibres of , and the product of rationally connected varieties is rationally connected.
Next, choose a curve which dominates . Deform and specialize until it breaks off a section. We have a map of moduli spaces
To make all of this more precise, we look at stable maps. Given a smooth projective variety over , and given and , we construct a space consisting of maps such that is a connected nodal curve of arithmetic genus , is a morphism, and , together with a stability criterion.
This space is projective, and given any morphism , one has an induced map from . If , then is a compactification of the space of branched covers. We want to show that is surjective. (We only need the coarse moduli space; in fact, at least in characteristic zero we have a Deligne-Mumford stack.)
When is , the bound is sharp. If is not algebraically closed, then pick a finite extension , and consider ; this has a polynomial of degree 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 smooth projective, and either for or for , then has a rational point. (These two are equivalent in characteristic zero.) The analog over function fields is false. In particular, there exists a family of Enriques surfaces over ( is a curve over ) which has no section.
The general statement: Let be a proper morphism of varieties over . Suppose that for all maps , there exists a pullback . It would suffice for there to exist dominating such that the general fibre of is rationally connected. It is a theorem that this is necessary and sufficient (G, Harris, Mazur, Starr).
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