Given a positive integer $d$, does there exist an integer $n$ that depends only on $d$ (or perhaps also on the dimension of $X$), such that for any degree $d$ finite étale covering $\pi: \widetilde X \to X$ of projective varieties and very ample line bundle $\mathcal L$ on $X$, $\pi^\ast(\mathcal L)^{\otimes n}$ is very ample.
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The answer is no. Take for $X$ a (smooth) plane curve of degree $2p+3$. There exists a line bundle $M$ on $X$ with $M^{2}=K_X$ and $h^0(M)=0$. Then $\eta :=M(-p)$ is a line bundle of order 2 in $JX$, giving rise to a double étale covering $\pi :\tilde{X}\rightarrow X $. Put $\mathscr{L}=\mathscr{O}_X(1)$. Then $$H^0(\tilde{X},\pi ^*\mathscr{L}^p)=\pi^* H^0(X,\mathscr{L}^p)\oplus \pi^*H^0(X,\mathscr{L}^p\otimes \eta )=\pi^*H^0(X,\mathscr{L}^p)\,.$$ This means that the map defined by $\pi ^*\mathscr{L}^p$ factors through $\pi $, hence $\pi ^*\mathscr{L}^p$ is not very ample.
