Blocking sets and plane curves over finite fields

Let 𝔽_q be a finite field, and let ℙ²(𝔽_q) denote the projective plane over 𝔽_q. A subset B ⊆ ℙ²(𝔽_q) is called a blocking set if it intersects every line defined over 𝔽_q. Given an algebraic plane curve C in ℙ², under what conditions does the set of 𝔽_q-rational 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 defined over 𝔽_q 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 𝔽_q-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!