# Pullback of very ample line bundles under finite étale covering

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.

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.