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Catalogue
  1. 1. rationality
  2. 2. Exercise
    1. 2.1. title

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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