Convex separably rationally connected complete intersections

Low Degree Complete Intersections Are Rationally Simply Connected

We prove smooth, low degree complete intersections in projective space are rationally simply connected in a strong sense. Using a result of Hassett, we deduce weak approximation for smooth, low degree complete intersections defined over the function field of a curve.

Convex Rationally 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.

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...

One cycles on rationally connected varieties

We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersections of index at least 2 is generated by lines...