Double coverings between smooth plane curves. (English) Zbl 1145.14027

Harui, Takeshi; Komeda, Jiryo; Ohbuchi, Akira

Kodai Math. J. 31, No. 2, 257-262 (2008).

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.

Reviewer: Cicero Carvalho (Uberlandia)

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