Recently, Graber, Harris and Starr proved that any family of rationally connected varieties over a smooth curve has a section. A complex projective variety (or manifold) M is rationally connected when every two points in M lie on a rational curve in M. In these lectures we will explain the strategy for proving an analogous result when the base of the family is a surface. We motivate one of the necessary definitions by an analogy with topology. In topology any fibration over a circle has a section when the fibres are (path) connected, and a fibration over a 2-sphere has a section when the fibres are 1-connected. We'd like to find a parallel definition of rationally 1-connected varieties. For example, projective spaces should be rationally 1-connected under any reasonable definition. Families of Brauer-Severi varieties over surfaces are counter-examples to a naive generalization of the Graber-Harris-Starr result, even though the fibres are all projective spaces. For such a family, it is well known that the obstruction for the existence of a section is a Brauer class on the base surface, i.e., it is a cohomological obstruction. The desired generalization of the Graber-Harris-Starr result is that for families of rationally 1-connected varieties over surfaces, the Brauer class is the only obstruction. The aim of the lectures will be to give an overview of the proof of this for a suitable class of families of varieties.
A rationally connected variety is very roughly a variety on which there lie a lot of rational curves, i.e., there are lots of nonconstant maps from P1 into the variety, in fact enough to connect any two points. By analogy with topology it seems reasonable to expect that (algebro-geometric) fibrations with rationally connected fibres over algebraic curves should have sections. This is a recent theorem by Graber, Harris and Starr. In this first lecture we would like to describe this result, give a hint of its proof and mention some related results such as Tsen's theorem and the period-index problem. Furthermore, we will describe a somewhat imprecise conjectural generalization of this theorem to families of varieties over surfaces. Finally, we will state a variant of this conjecture that forms the main theorem that will be discussed in Lectures 2-5.
In this lecture we describe some results on the rational connectivity of the moduli spaces of rational curves on hypersurfaces of low degree and homogenous varieties. Families of homogenous varieties will give interesting cases where the main theorem applies. Families of hypersurfaces of low degree provide nontrivial examples where we (as yet) don't know how the conjectural framework applies.
Here we give a precise formulation of the main theorem and we outline the proof, in particular reducing the proof to a statement on families of varieties over curves.
We describe how to get the energetic morphisms that are used in the proof of the main theorem.
We put everything together.