## Looking for rational curves on rationally connected varieties

### Zhiyu Tian

The study of local-global principles for rationally connected varieties over function fields naturally leads to the following geometric question: given a rationally connected variety with a cyclic group action, and two fixed points, can we find an equivariant rational curve which connects the two points? I will present some positive results (joint work with R. Zong) and some negative results (Jason Starr).