An arc in the projective plane over a finite field Fq is a collection of points, no three of which lie on a line. Segre’s theorem tells us that the largest size of an arc is q+1 when q is odd and q+2 when q in even. Moreover, it classifies these maximal arcs when q is odd, stating that every such arc is the set of rational points of a smooth conic.
We will give an overview of problems about arcs in the plane and in higher dimensional projective spaces. Our goal will be to use algebraic techniques to try to understand these extremal combinatorial configurations. We will also see connections to special families of error-correcting codes and to modular forms.