I would like to find an example of very ample line bundle on a smooth projective variety whose pull-back under an étale covering is non-very ample. More precisely:
Is there an example of very ample line bundle $L$ on a smooth projective variety $Y$ such that the pullback $f^\ast L$ under a finite étale morphism $f\colon X\rightarrow Y$ is non-very ample?
In the above situation, $f^\ast L$ is ample and globally generated. Moreover, given that the linear series $f^\ast|L|\subseteq |f^\ast L|$ separates tangent vectors, it is clear that $|f^\ast L|$ separates tangents as well. One has to check that $|f^\ast L|$ does not separate points.
My attempts regarded surfaces, but I did not find anything.
Any help is greatly appreciated.
Best Answer
Let $f \colon X \to Y$ be am etale double covering of a general plane sextic curve and let $L$ be the restriction of $\mathcal{O}(1)$ from the plane. Then $$ H^0(X, f^*L) \cong H^0(Y, f_*f^*L) \cong H^0(Y, L) \oplus H^0(Y, L \otimes \xi), $$ where $\xi$ is a line bundle of order 2 (i.e., such that $\xi^2 \cong \mathcal{O}_Y$) corresponding to the double covering. So, it is enough to check that $H^0(Y, L \otimes \xi) = 0$.
First, note that if $D$ is an effective divisor corresponding to a global section of $L \otimes \xi$ then $2D$ corresponds to $L^2$, hence it is cut out on $Y$ by a conic on the plane. Since it has the form $2D$, the conic is everywhere tangent to $Y$. Thus, it is enough to check that a general plane sextic does not have everywhere tangent conics.
Consider the variety $M$ of triples $(Y,C,D)$, where $Y$ is a smooth plane sextic, $C$ is a conic and $D$ is a divisor (of degree 6) in $Y$ such that $Y \cap C = 2D$. Note that $D$ is also a divisor on $C$. Moreover, the space $N$ of pairs $(C,D)$ has dimension $5 + 6 = 11$. Moreover, since plane sextics cut out a complete linear system (of dimension 12) on $C$, it follows that the fiber of the projection $M \to N$ has codimension 12 in the space of all sextic curves. Therefore, the dimension of $M$ is less by 1 than the dimension of the space of all plane sextics, hence the projection of $M$ to this space is not dominant, hence a general $Y$ is not in the image.