Points on singular Frobenius nonclassical curves (Q2400473)

Let \(k\) be the finite field of order \(q\). The Stöhr-Voloch theory [\textit{K.-O. Stöhr} and \textit{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 \textit{\(q\)-Frobenius nonclassical}. In this case, if \(\mathcal X\) is plane of degree \(d\), \textit{A. Hefez} and \textit{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.