Arcs in the Projective Plane

Nathan Kaplan
Yale University

A $(k,n)$-arc of $\mathbb{P}^2(\mathbb{F}_q)$ is a set of $n$ rational points in the projective plane, no more than $k$ of which lie on a line. Segre's famous theorem on $(2,n)$-arcs says that when $q$ is odd, the largest $(2,n)$-arc is of size $q+1$ and every such arc is the zero set of a smooth conic. When $q$ is even there are $(2,q+2)$-arcs called hyperovals, which are yet to be completely classified.

In addition to asking for the largest size of an arc, we can ask for the number of arcs of a given size. We will explain how counting arcs with a small number of points is related to counting del Pezzo surfaces with many rational points. We will also discuss approaches to constructing large $(k,n)$-arcs based on properties of algebraic curves over finite fields.

Presentation (PDF File)

Back to Workshop IV: Finding Algebraic Structures in Extremal Combinatorial Configurations