Frobenius indices of certain curves over finite fields (Q1566947)

The author is concerned with an algebraic curve \(X \subset \mathbb{P}^N\) \((N\geq 3)\) over a finite field of characteristic \(p>0\) lying in an \(N\)-dimensional projective space \(\mathbb{P}^N\) which possesses an order-sequence in the sense of \textit{A. Garcia} and \textit{M. Homma} [in: Proc. Conf. Inst. Experimental Math., Essen 1992, 27-41 (1994; Zbl 0824.14019)] or of \textit{K.-O. Stöhr} and \textit{J. F. Voloch} [Proc. Lond. Math. Soc., III. Ser. 52, 1-19 (1986; Zbl 0593.14020)]. Considering such a curve \(X \subset \mathbb{P}^N\) over a finite field \(\mathbb{F}_{q'}\) with \(q'\) elements (\(q'\) is a power of \(p\)), it possesses the \(q'\)-Frobenius order-sequence in the sense of the papers cited above, too. Let \(x_0: x_1: x_2:\cdots: x_N\) be the coordinate functions, \(k(X)\) be the function-field of \(X \subset \mathbb{P}^N\), and \(\{D^{(r)}\); \(0\leq r\in \mathbb{Z}\}\) be the system of Hasse-Schmidt derivatives on \(k(X)\). Put \({\mathfrak x}= (x_0, x_1, x_2,\dots, x_N)\). Then each of both sequences which are denoted by \(0= \varepsilon_0< \varepsilon_1< \varepsilon_2<\dots< \varepsilon_N\) (order-sequence) and \(0= \nu_0< \nu_1<\dots< \nu_N\) (\(q'\)-Frobenius order-sequence), means the minimal sequence consisting of integers, in the lexicographic order, such that the \(N+1\) row-vectors \(D^{(\varepsilon_i)} \cdot {\mathfrak x}\) \((0\leq i\leq N)\) are linearly independent over \(k(X)\), the \(N+1\) row-vectors \({\mathfrak x}^{q'}\), \(D^{(\nu_i)} \cdot{\mathfrak x}\) \((0\leq i\leq N-1)\) are so, respectively. By proposition 2.1 in the paper by \textit{K.-O. Stöhr} and \textit{J. P. Voloch} cited above, for the relationship between both sequences, it is known that there exists an integer \(I\) depending on \(q'\) \((1\leq I\leq N)\) such that \(\nu_i= \varepsilon_i\) (whenever \(i<I\)), \(\varepsilon_{i+1}\) (whenever \(i\geq I\)). This value \(I\) is named Frobenius index. The author denotes by \(\iota(q';X)\) this value for the curve \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\). Now, let an integer \(N\geq 3\) and an odd prime number \(p\) be arbitrarily given. Take arbitrarily an integer \(I\) \((1\leq I\leq N)\). Then the author gives an example of \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\) satisfying \(\iota (q';x)= I\), where \(q'\) is some power of \(p\). This example is obtained by a complete intersection in \(\mathbb{P}^N\) of \(N-I\) Fermat equations and \(I-1\) Artin-Schreier equations. Concerning this, the author notes that (a): in case \(N=3\), an example of \(X \subset \mathbb{P}^3\) with ``\(\iota (q'X)= 1\) for some \(q'\)'' is given in the Garcia-Homma paper (see above) or by \textit{A. Garcia} and \textit{J. F. Voloch} [Bol. Soc. Bras. Mat., Nova Sér. 21, No. 2, 159-175 (1991; Zbl 0766.14021)] which is a complete intersection of Fermat equations. The author's example for \(\iota (q';X)= 1\) in \(N=3\) in the same one as (a). (b): For any \(N\), an example of \(X \subset \mathbb{P}^N\) with ``\(\iota (q';X)= N\) for any \(q'\)'' which is an image of \(\mathbb{P}^1\) is given in the Garcia-Homma paper cited above. (c): For any \(N\), an example of \(X \subset \mathbb{P}^N\) with ``\(\iota (q';X)= N-1\) for some \(q'\)'' which is an image of \(\mathbb{P}^1\) is given in the same paper by Garica and Hamma. In the case of the author's example for \(\iota (q';X)= 1\), since it is seen that \(X\) is smooth, the author calculates the number of \(\mathbb{F}_{q'}\)-rational points on the curve \(X\), by applying the formula of theorem 1 in a paper by \textit{A. Hefez} and \textit{J. F. Voloch} [Arch. Math. 54, No. 3, 263-273 (1990; Zbl 0662.14016)]. Let \(q= p^e\), \(q'= q^2\), where \(e\) is a positive integer satisfying \(N\leq p^e\). Consider the curve \(X \subset \mathbb{P}^N\) over \(\mathbb{F}_{q'}\) of the author's example for \(\iota (q';X)= 1\). Then the genus of \(X\) equals \(1+ \frac 12 (q+1)^{N-1} [(N-1) q-2]\) and moreover \(\# X(\mathbb{F}_{q'})\) equals \((q+1)^{N-1} [q^2+ 1- (N-1)q]\).