An n-arc in the k-dimensional projective space over a finite field F_q is a collection of n points with no k+1 in a hyperplane.  What is the largest size of an n-arc?  The MDS Conjecture proposes an answer to this question for all k and q.

For k = 2 this question asks for the maximum number of points in the projective plane over F_q with no 3 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 see interesting examples of algebraic varieties in positive characteristic and see connections to the theory of error-correcting codes.