Approximately Rationally or Elliptically Connected Varieties

Rationally connected varieties

The aim of these notes is to provide an introduction to the theory of rationally connected varieties, as well as to discuss a recent result by T. Graber, J. Harris and J. Starr.

Convex Rationally Connected Varieties

Rationally Connected Varieties

1 13 4. Free rational curves 14 5. Uniruledness 16 5.1. Definitions equivalent to uniruledness 16 5.2. Examples and consequences of uniruledness 18 6. Rational connectivity 20 6.1. Definitions equivalent to rational connectivity 21 6.2. Rational chain-connectivity implies rational connectivity 22 6.3. Examples and consequences of rational connectivity 26 7. The MRC quotient 27 7.1. Quotients by...

Families of Rationally Connected Varieties

1.1. Statement of results. We will work throughout over the complex numbers, so that the results here apply over any algebraically closed field of characteristic 0. Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X . This is clearly a birationally invariant property. When X is smooth, this turns out to ...

Rationally Connected Varieties over Local Fields

LetX be a proper variety defined over a fieldK. Following [Manin72], two points x, x ∈ X(K) are called R-equivalent if they can be connected by a chain of rational curves defined over K, cf. (4.1). In essence, two points are R-equivalent if they are “obviously” rationally equivalent. Several authors have proved finiteness results over local and global fields (cubic hypersurfaces [Manin72, Swinn...