For instance, in these notes of a talk by Martin Sombra http://www.maia.ub.edu/~sombra/talks/mega2003/mega2003.pdf
A concept of height is defined for projective varieties defined over $\mathbb{Q}$, see pages 7-8 in the link attached.
Basically, $[k]:\mathbb{P}^{n}\to \mathbb{P}^{n}$ is defined via $(x_0:\dots:x_n)\mapsto(x_0^k:\dots:x_n^k).$ For a variety $X\subset \mathbb{P}^n$ defined over $\mathbb{Q}$ they set $h_{naive}(X)=h(Ch_X)$ where $Ch_X$ is a primitive form of the chow form of $X$, and the set:
$h(X)=deg(X)\cdot \lim_{k\to\infty}\frac{h_{naive}([k]X)}{kdeg([k]X)}$
Similar notation can be found in papers by Clarice philipon and his students.
My question is: What is meant exactly by $[k]X$? It looks like set theoretic image but that may not end up an algebraic set even. For example if $X=\{xz=y^{2}\}$ then $[k](X)=\{x^{1/k}z^{1/k}=y^{2/k}\}$ which is not algebraic. Maybe inverse image is meant?
In any case I would be happy for your advice. Thank you!