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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.

14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H45 Special algebraic curves and curves of low genus
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