We know how to define the height on projective space. This height does very important role to find finiteness of rational points. If we have a morphism between projective space, then it works good with the height. For example, we can define the canonical height with the given morphism. The canonical height on elliptic curve is one of the good example.

However, if we have rational map (which has some points where given map is not defined, like 1/x), we can't derive the same result since the rational map is not a functor. To do the similar thing, we should expand the domain to make give map morphism first and should define the height of expanded domain. I will show two ways of constructing height of give varieties and related facts.