Uniruled and Rationally Connected Varieties

Abstract

As we saw in Section 3.2, there exists a rational curve through every point of a Fano variety. A variety X with this property is called uniruled: there exists on X a rational curve whose deformations cover a dense open subset of X. We show that most other versions of this definition that come to mind turn out to be equivalent, at least over an uncountable algebraically closed field: the thing that one wants to rule out is a variety not covered by rational curves say of fixed degree (with respect to some ample divisor), but which is still a (countable) union of rational curves (the degrees going to infinity).