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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.

##### MSC:
 14H45 Special algebraic curves and curves of low genus 14Hxx Curves in algebraic geometry
##### Keywords:
algebraic curve; Frobenius nonclassical curve; finite field
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##### References:
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