On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola. (English) Zbl 1465.11158
Summary: Let \(\mathcal{G}\) be the projective plane curve defined over \(\mathbb{F}_q\) given by \[a X^n Y^n - X^n Z^n - Y^n Z^n + b Z^{2 n} = 0,\] where \(a b \notin \{0, 1 \}\), and for each \(s \in \{2, \ldots, n - 1 \}\), let \(\mathcal{D}_s^{P_1, P_2}\) be the base-point-free linear series cut out on \(\mathcal{G}\) by the linear system of all curves of degree \(s\) passing through the singular points \(P_1 = (1 : 0 : 0)\) and \(P_2 = (0 : 1 : 0)\) of \(\mathcal{G}\). The present work determines an upper bound for the number \(N_q(\mathcal{G})\) of \(\mathbb{F}_q\)-rational points on the nonsingular model of \(\mathcal{G}\) in cases where \(\mathcal{D}_s^{P_1, P_2}\) is \(\mathbb{F}_q\)-Frobenius classical. As a consequence, when \(\mathbb{F}_q\) is a prime field, the bound obtained for \(N_q(\mathcal{G})\) improves in several cases the known bounds for the number \(n_P\) of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point \(P\) distinct from its vertices.
MSC:
| 11G20 | Curves over finite and local fields |
| 14G15 | Finite ground fields in algebraic geometry |
| 14G05 | Rational points |
| 51E15 | Finite affine and projective planes (geometric aspects) |
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