JournalsrsmupVol. 135pp. 39–61

Projection of a nonsingular plane quintic curve and the dihedral group of order eight

  • Takeshi Takahashi

    Niigata University, Japan
Projection of a nonsingular plane quintic curve and the dihedral group of order eight cover
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Abstract

Let CC be a nonsingular plane quintic curve over the complex number field C\mathbb{C}, and let πP ⁣:CP1\pi_P\colon C \rightarrow \mathbb{P}^1 be a projection from PCP \in C. Let LPL_P be the Galois closure of the field extension C(C)/C(P1)\mathbb{C}(C)/\mathbb{C}(\mathbb{P}^1) induced by πP\pi_P, where C(C)\mathbb{C}(C) and C(P1)\mathbb{C}(\mathbb{P}^1) are the rational function fields of CC and P1\mathbb{P}^1, respectively. We call the point PP a D4D_4-point if the Galois group of LP/C(P1)L_P/\mathbb{C}(\mathbb{P}^1) is isomorphic to the dihedral group D4D_4 of order eight. In this paper, we prove that the number of D4D_4-points for CC equals 00, 11, 33, 55, or 1515, and show that the curve with 1515 D4D_4-points is projectively equivalent to the Fermat quintic curve.

Cite this article

Takeshi Takahashi, Projection of a nonsingular plane quintic curve and the dihedral group of order eight. Rend. Sem. Mat. Univ. Padova 135 (2016), pp. 39–61

DOI 10.4171/RSMUP/135-3