Definition 2.39 works better when the variety $X$ is projective. In this case, for every cycle $Y$ there exist sections that meet $Y$ properly, thanks to the moving lemma. This does not necessarily occur for arbitrary proper varieties. Nevertheless, we will be able to define the global height (Definition 2.56) of any cycle of a proper variety by using Chow’s lemma. Similarly we will be able to define the toric local height (Definition 6.1) of any cycle of a proper toric variety.