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Double coverings between smooth plane curves. (English) Zbl 1145.14027
In this paper the authors classify the pairs of smooth plane curves $$(C, C')$$ such that there exists a covering $$\pi: C \rightarrow C'$$ of degree 2. The main result is as follows.
Theorem. Let $$C$$ and $$C'$$ be two smooth plane curves of degree $$d$$ and $$d'$$, respectively. Then there exists no double covering from $$C$$ to $$C'$$, except for the following cases:
(i) $$C'$$ is rational $$(d'\leq 2$$) and $$C$$ is rational or elliptic ($$d \leq 3$$);
(ii) $$C$$ and $$C'$$ are elliptic ($$d = d'= 3$$);
(iii) $$C'$$ is elliptic and $$C$$ is a bielliptic plane quartic ($$d'= 3$$, $$d = 4$$).
In particular, no smooth plane curve can be a double covering of a smooth plane curve of degree greater than 3.

##### MSC:
 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14H45 Special algebraic curves and curves of low genus
##### Keywords:
double covering of curves; plane curves; elliptic curves
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