Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

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)
  • Received : 2015.10.02
  • Published : 2017.01.01
Download PDF
⟨ Previous Next ⟩

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 ${\varphi}$.

Keywords

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