Blocking sets and plane curves over finite fields

Let Fq be a finite field of order q. A subset B of a projective plane over Fq is called a blocking set if every line defined over Fq intersects B. Given an algebraic plane curve C, under what conditions does the set of Fq-points on C form a blocking set? We show that curves of low degree do not yield blocking sets. As an example of this principle, cubic plane curves do not give rise to blocking sets whenever q is at least 5. In contrast, we provide explicit constructions of smooth plane curves of sufficiently large degree whose points do form blocking sets. Furthermore, in a precise asymptotic sense, most plane curves over finite fields are not blocking. Determining the minimum degree of a blocking plane curve over a given finite field remains an open problem. This is joint work with Dragos Ghioca and Chi Hoi Yip.

Shamil Asgarli received his B.Sc. in Mathematics from UBC in 2014 and a Ph.D. from Brown University in 2019. He then returned to UBC as a postdoctoral fellow from 2019 to 2022. The first two math courses he ever took, Math 200 (Multivariable Calculus) and Math 223 (Honours Linear Algebra), were both taught in the very same Mathematics Building whose 100th anniversary we are now celebrating!