# RELATING GALOIS POINTS TO WEAK GALOIS WEIERSTRASS POINTS THROUGH DOUBLE COVERINGS OF CURVES

• Komeda, Jiryo (Department of Mathematics Kanagawa Institute of Technology) ;
• Takahashi, Takeshi (Department of Information Engineering Faculty of Engineering Niigata University)
• Published : 2017.01.01

#### Abstract

The point $P{\in}{\mathbb{P}}^2$ is referred to as a Galois point for a nonsingular plane algebraic curve C if the projection ${\pi}_P:C{\rightarrow}{\mathbb{P}}^1$ from P is a Galois covering. In contrast, the point $P^{\prime}{\in}C^{\prime}$ is referred to as a weak Galois Weierstrass point of a nonsingular algebraic curve C' if P' is a Weierstrass point of C' and a total ramification point of some Galois covering $f:C^{\prime}{\rightarrow}{\mathbb{P}}^1$. In this paper, we discuss the following phenomena. For a nonsingular plane curve C with a Galois point P and a double covering ${\varphi}:C{\rightarrow}C^{\prime}$, if there exists a common ramification point of ${\pi}_P$ and ${\varphi}$, then there exists a weak Galois Weierstrass point $P^{\prime}{\in}C^{\prime}$ with its Weierstrass semigroup such that H(P') = or , which is a semigroup generated by two positive integers r and 2r + 1 or 2r - 1, such that P' is a branch point of ${\varphi}$. Conversely, for a weak Galois Weierstrass point $P^{\prime}{\in}C^{\prime}$ with H(P') = or , there exists a nonsingular plane curve C with a Galois point P and a double covering ${\varphi}:C{\rightarrow}C^{\prime}$ such that P' is a branch point of ${\varphi}$.

#### Acknowledgement

Supported by : JSPS

#### References

1. E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 267. Springer-Verlag, New York, 1985.
2. H. C. Chang, On plane algebraic curves, Chinese J. Math. 6 (1978), no. 2, 185-189.
3. M. Coppens and T. Kato, The Weierstrass gap sequence at an inflection point on a nodal plane curve, aligned inflection points on plane curves, Boll. Un. Mat. Ital. B. (7) 11 (1997), no. 1, 1-33.
4. S. Fukasawa, Galois points for a plane curve in arbitrary characteristic, Geom. Dedicata 139 (2009), 211-218. https://doi.org/10.1007/s10711-008-9325-2
5. S. Fukasawa, On the number of Galois points for a plane curve in positive characteristic. III, Geom. Dedicata 146 (2010), 9-20. https://doi.org/10.1007/s10711-009-9422-x
6. T. Kato, On the order of a zero of the theta function, Kodai Math. Sem. Rep. 28 (1976/77), no. 4, 390-407. https://doi.org/10.2996/kmj/1138847520
7. S. J. Kim and J. Komeda, Numerical semigroups which cannot be realized as semigroups of Galois Weierstrass points, Arch. Math. (Basel) 76 (2001), no. 4, 265-273. https://doi.org/10.1007/s000130050568
8. S. J. Kim and J. Komeda, The Weierstrass semigroup of a pair of Galois Weierstrass points with prime degree on a curve, Bull. Braz. Math. Soc. (N.S.) 36 (2005), no. 1, 127-142. https://doi.org/10.1007/s00574-005-0032-4
9. S. J. Kim and J. Komeda, The Weierstrass semigroups on the quotient curve of a plane curve of degree ${\leq}$ 7 by an involution, J. Algebra 322 (2009), no. 1, 137-152. https://doi.org/10.1016/j.jalgebra.2009.03.023
10. K. Miura and H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra 226 (2000), no. 1, 283-294. https://doi.org/10.1006/jabr.1999.8173
11. K. Miura and H. Yoshihara, Field theory for the function field of the quintic Fermat curve, Comm. Algebra 28 (2000), no. 4, 1979-1988. https://doi.org/10.1080/00927870008826940
12. I. Morrison and H. Pinkham, Galois Weierstrass points and Hurwitz characters, Ann. Math. (2) 124 (1986), no. 3, 591-625. https://doi.org/10.2307/2007094
13. F. Torres, Weierstrass points and double coverings of curves. With application: symmetric numerical semigroups which cannot be realized as Weierstrass semigroups, Manuscripta Math. 83 (1994), no. 1, 39-58. https://doi.org/10.1007/BF02567599
14. H. Yoshihara, Function field theory of plane curves by dual curves, J. Algebra 239 (2001), no. 1, 340-355. https://doi.org/10.1006/jabr.2000.8675