Together with Marco Timpanella (University of Perugia) and Giovanni Zini (University of Modena and Reggio Emilia), the research group organize monthly seminars.
20 November 2025 [4.00 pm - Italian time] - Joao Paulo Guardieiro Sousa (Federal University of Uberlândia, Brazil)
Title: Frobenius nonclassical curves
Abstract: In 1986, Stöhr and Voloch were interested in studying the geometry of algebraic curves defined over Fq. As a consequence of their work, they presented a bound on the number of rational points for a curve. This bound not only proves again Hasse-Weil's bound, but also allows its improvement under some circumstances. Their work also divided algebraic curves into two types: the Frobenius classical curves and the Frobenius nonclassical ones.
Frobenius nonclassical curves became an interesting topic of research since they have many rational points and some relevant geometric properties. As a consequence, the work of Stöhr and Voloch made possible a better understanding of maximal curves, the study of optimal curves, the study of sections of surfaces, the determination of the automorphism group of curves and advances on Finite Geometry theory. More recently, the definition of Frobenius (non)classical curves was generalized to surfaces. This generalization highly benefits from the study of curves and was already used to, for instance, study arithmetic properties of curves lying on surfaces, characterize Hermitian surfaces, and study blocking sets.
In this talk, I will present some results on Frobenius nonclassical curves, and their connection to minimal value set polynomials and the problem of finding complete arcs. I will also present our results on the classification of Frobenius nonclassical trinomial and quadrinomial curves. This is a joint work with Herivelto Borges and Tiago Aprígio.