Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.

Is there a constant $C=C(\mathrm{deg}(f))$ such that $Cf^*H$ is very ample? If the answer is in general no, are there natural conditions to impose?

Let $X$ and $Y$ be normal projective varieties over $\mathbb{C}$ of dimension $n$. Let $f: X \rightarrow Y$ be a finite morphism. Also, let $H$ be a very ample divisor on $Y$.

Is there a constant $C=C(\mathrm{deg}(f),n)$ depending just on degree and dimension such that $Cf^*H$ is very ample? If the answer is in general no, are there natural conditions to impose?