Blocking plane curves over finite fields

Let F_q be a finite field, and consider the set P^2(F_q) of all F_q-points in the projective plane. Originating from game theory, a subset B of P^2(F_q) is called a blocking set if B meets every line defined over F_q. Algebraic curves, especially those defined by Redei-type polynomials, are powerful in studying blocking sets. One can reverse the engine and ask the following question: given an irreducible (or smooth) plane curve C in P^2, when does C(F_q) form a nontrivial blocking set? Alternatively, given d and q, does there exist an irreducible (or smooth) curve C with degree d defined over F_q that give rise to a nontrivial blocking set?

In this talk, I will give a partial answer to these questions using a mixture of tools from arithmetic geometry, arithmetic statistics, incidence geometry, and number theory. This is a joint work with Shamil Asgarli and Dragos Ghioca.

Time: 3:00-4:00pm

Location: MATH 126

Seminar Website: https://yifeng-huang-math.github.io/seminar_ubc_ag_22f.html