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Points on singular Frobenius nonclassical curves. (English) Zbl 1374.14027
Let \(k\) be the finite field of order \(q\). The Stöhr-Voloch theory [K.-O. Stöhr and J. F. Voloch, Proc. Lond. Math. Soc. (3) 52, 1–19 (1986; Zbl 0593.14020)] allows us to bound the number \(N\) of \(k\)-rational points of a nondegenerate, nonsingular, projective, geometrically irreducible algebraic \(\mathcal X\subseteq {\mathbb P}^t(\bar k)\) defined over \(k\). Thus \(N\leq \text{deg}(S)\) where \(S\) is certain \(k\)-divisor on \(\mathcal X\) which roughly speaking depends on the points \(P\in \mathcal X\) such that \(\phi(P)\) belongs to the osculating hyperplane \(T_P{\mathcal X}\) at \(P\) \((*)\), where \(\phi\) is the Frobenious morphism on \(\mathcal X\) (induced by \(x\mapsto x^q\)). We notice that \(N\) is also the number of fixed points of \(\Phi\). If \((*)\) holds for almost all \(P\), the curve is called \(q\)-Frobenius nonclassical. In this case, if \(\mathcal X\) is plane of degree \(d\), A. Hefez and J. F. Voloch noticed that \(N=d(q-d+2)\) [Arch. Math. 54, 263–273 (Zbl 0662.14016)].
In the paper under review, the authors show that \(N^*\geq d(q-d+2)\) for any plane curve \(\mathcal X\) of degree \(d\) Frobenius nonclassical over \(k\) (here property \((*)\) must holds for nonsingular points), where \(N^*\) is the number of \(k\)-rational points of the nonsingular model of \(\mathcal X\). Naturally \(N^*=d(q-d+2)\) only if \(\mathcal X\) is nonsingular.

14H45 Special algebraic curves and curves of low genus
14Hxx Curves in algebraic geometry
Full Text: DOI arXiv
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[3] Hefez, A; Voloch, JF, Frobenius nonclassical curves, Arch. Math., 54, 263-273, (1990) · Zbl 0662.14016
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