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rationality

\(rational \subset unirational \subset rationally connected \subset uniruled\)

where uniruled if \(\mathbb{P}^1\times Y \dashrightarrow X\) dominate.
Def: A variety \(X\) is ratinally connected if it is proper and exists a variety \(M\) and a rational map
$$ e:\mathbb{P}^1\times M \dashrightarrow X $$
s.t. the rational map
$$ \mathbb{P}^1\times \mathbb{P}^1 \times M \dashrightarrow X\times X $$
is dominant

An example for uniruled but not rationality connected:
\(X = \mathbb{p}^1\times E\) with \(g(E)\geqslant 1\). All rational curves in \(X\) is vertical.

Exercise

Let \(f: Y \dashrightarrow X\) be a rational dominant map, and \(Y,X\) smooth projective varities, then \(\kappa (Y) \geqslant \kappa (X)\)

Remark:

1. NOT true for $char k >0 $
2. \(X\) is uniruled, then $\kappa (X)= - \infty $
3. \(A\) abelian variety over \(\mathbb{C}\), then \(\kappa (A)=0\)

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