Double covers of plane curves of degree six with almost total flexes. (English) Zbl 1427.14065

Kim, Seon Jeong; Komeda, Jiryo

Bull. Korean Math. Soc. 56, No. 5, 1159-1186 (2019).

Let \(f: X\to C\) be a double covering of smooth projective curves. Let \(P\in C\) be a ramification point of \(f\) with \(f(P')=P\). Let \(H(P)\) and \(H(P')\) be the Weierstrass semigroup of \(P\) and \(P'\), respectively. The authors pose the problem of the clasification of all \(H(P')\) for all smooth plane curves. They solved it here when \(C\) has degree \(6\). They did the case \(\deg (C)=5\) in [S. J. Kim and J. Komeda, Kodai Math. J. 38, No. 2, 270–288 (2015; Zbl 1327.14159)] and did some other cases in [S. J. Kim and J. Komeda, Bull. Korean Math. Soc. 55, No. 2, 611–624 (2018; Zbl 1401.14158)].

Reviewer: Edoardo Ballico (Povo)

MSC:

14H55	Riemann surfaces; Weierstrass points; gap sequences
14H50	Plane and space curves
14H30	Coverings of curves, fundamental group
20M14	Commutative semigroups

Keywords:

numerical semigroup; Weierstrass semigroup of a point; double cover of a curve; plane curve of degree 6

References:

[1]	T. Harui and J. Komeda, Numerical semigroups of genus eight and double coverings of curves of genus three, Semigroup Forum 89 (2014), no. 3, 571-581. https://doi.org/ 10.1007/s00233-014-9590-3 · Zbl 1309.14026
[2]	S. J. Kim and J. Komeda, Weierstrass semigroups on double covers of genus 4 curves, J. Algebra 405 (2014), 142-167. https://doi.org/10.1016/j.jalgebra.2014.02.006 · Zbl 1395.14028
[3]	, Weierstrass semigroups on double covers of plane curves of degree 5, Kodai Math. J. 38 (2015), no. 2, 270-288. https://doi.org/10.2996/kmj/1436403890 · Zbl 1327.14159
[4]	, Weierstrass semigroups on double covers of plane curves of degree six with total flexes, Bull. Korean Math. Soc. 55 (2018), no. 2, 611-624. · Zbl 1401.14158