Transformation of height on projective varieties

Question

I am trying to find a reference for the following statement: let $X$ and $Y$ be projective varieties defined over $ \mathbb{Q}$ and $\phi: X \to Y$ be a rational map defined over $ \mathbb{Q}$. Denote the height on $X$ and $Y$ by $H$. Then there exists $C,d >0$ such that $$H(\phi(p) ) \le C \phi(p)^d $$ holds for any rational point on $X$.

I can find various statements of this kind in the literature in which $X$ and $Y$ are both projective spaces and $\phi$ is a morphism, but could not find the above statement. I would very appreciate help in finding a reference.

Answer 1

First, your statement can't be true as you've stated it, because you say it holds for any (by which I assume you mean all) rational points on $X$. But your map is a rational map, so there will be some points where your map is not defined! Aside from that, your inequality is correct on all algebraic ponts. This follows more-or-less by the triangle inquality. For maps $\mathbb{P}^m\dashrightarrow\mathbb{P}^n$, this result is Theorem B.2.5 in [1]. Then you can just cover your varieties by open sets on which your map extends to a rational map on projective space whose indeterminacy loci matche the indeterminacy locus of your map.

The opposite inequality is only valid for morphisms if you want $d\ge1$; but it will in fact be true off of a Zariski closed set if you take $d>0$ to be sufficiently small. This is in [2].

[1] Hindry, Marc; Silverman, Joseph H. Diophantine geometry. An introduction. Graduate Texts in Mathematics, 201. Springer-Verlag, New York, 2000.

[2] Silverman, J.H. Height estimates for equidimensional dominant rational maps. J. Ramanujan Math. Soc. 26 (2011), no. 2, 145–163. MR2816785