We start with the diagram:
Definition. The -rational points of satisfy weak approximation if for each a finite set of places, with completions , and open sets , there exists an with for each .
Note that for nonarchimedean places, , , then we have reduction maps
Remarks. By Hensel's lemma, gives a point in if is smooth.
If is regular, then if comes from a point in , then is regular.
Function field analog
Now consider the diagram
Definition. satisfies weak approximation if for any such set of data there exists so that .
Rationally connected case
Let be rationally connected, with model . Here we have the theorem:
Theorem. [Graber, Harris, Starr; Kollár] There exists a section . Choose points such that the fibres are smooth, and choose points ; then there exists a section with .
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 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 is rationally chain connected but is not itself rationally connected.
Problem. Let be a smooth projective variety, , a curve. If is rationally connected, show that satisfies weak approximation.
Problem. Given , of multiplicity one, does there exist an effective curve class such that , and .
Let be a projective smooth variety over . We have , the Néron-Severi group, and , the -cycles. We have the cone , the cone of effective divisors; we also have the cone of moving curves , consisting of cycle classes such that is irreducible and passes through the generic point of .
Given an effective divisor and a moving class , then .
Note that , the dual cone.
Theorem. [Demailly, Peternell] Equality holds, .
As an application, this allows us to find with the desired intersection properties.
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