Arcs in the projective plane and del Pezzo surfaces over finite fields

An $n$-arc in ${\bf P}^2({\Bbb F}_q)$ is a collection of n distinct points, no three of which lie on a line. A theorem of Segre says that when $q$ is odd the largest arc is of size $q+1$ and when $q$ is even the largest arc is of size $q+2$. In addition to asking for the largest size of an arc, we can ask for the number of arcs of a given size.

A del Pezzo surface of degree $d$ over ${\Bbb F}_q$ has at most $q^2 + (10 − d)q + 1\ {\Bbb F}_q$-rational points. A surface attaining this maximum is called split, and if all of these rational points lie on the exceptional curves of the surface then it is called full. We explain the connection between counting problems for arcs and the classification of these extremal del Pezzo surfaces, focusing on the case of del Pezzo surfaces of degree 3, cubic surfaces, and of degree 2, double covers of ${\bf P}^2$ branched over a quartic curve.

Nathan Kaplan