Arithmetic Questions Related to Rationally Connected Varieties

Fano 4-folds, flips, and blow-ups of points

“…Since Y i is rationally connected, by [GHS03] there exists C i ⊂ Y i section of f i , so that F i · C i = 1. Now let F 2 ⊂ Y 1 be the transform of F 2 .…”

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

“…Briefly, a k-point of Sections(X/S/B) is a pair (b, σ), where b ∈ B(k), and σ : S b → X b is a section of the restriction of X → S to S b . Applying [GHS03] to the morphism Sections(X/S/B) → B we see that it would suffice to produce an irreducible subvariety Z ⊂ Sections(X/S/B) such that the general fibre of Z → B is rationally connected.…”

“…In the paper [GHS03] it was shown that a family of birationally rationally connected varieties over a curve has a rational section. A variety X is called birationally rationally connected if a general pair of points (x, y) ∈ X × X can be connected by a rational curve, i.e., if there exists an open U ⊂ P 1 and a morphism f : U → X such that x, y ∈ f (U ).…”

“…How to generalize the result of [GHS03] to families of varieties over a surface? By analogy with topology one could hope for a statement of the following type: A family of (birationally) rationally simply connected varieties over a surface has a rational section.…”

“…For example, trying to use a property such as (1) above to prove (P) one runs into the problem of not having a result like [GHS03] for families of ind-schemes.…”