Mathematics Graduate Student Colloquium

MGSC Website: http://math.uci.edu/~mgsc/

Counting n-Arcs in Projective Planes

Kelly Isham
Thursday, May 9, 2019
4:00 pm - 4:50 pm
340N

Talk Abstract:

An n-arc in P^2(F_q) is a collection of n distinct points such that no three lie on a line. In 1988, Glynn gave an algorithm for counting the number of n-arcs in a projective plane of order q. Previous work has shown that for n less than 10, the number of n-arcs is polynomial or quasipolynomial (i.e. the formula is given by a finite number of polynomials depending on residue classes). This leads to the question - will the number of n-arcs over P^2(F_q) continue to be quasipolynomial in q? In this talk, we discuss a modification of Glynn's algorithm that makes computation simpler and we explain how the problem of counting the number of n-arcs in the projective plane over F_q is equivalent to counting the number of rational points on certain varieties. We use this new approach to prove that the number of 10-arcs in a projective plane over F_q is not quasipolynomial. We discuss analagous results for larger n and we relate this counting problem to the study of F_q-points on Grassmannians and to MDS codes.

About the Speaker:

Kelly is a 3rd year PhD student interested in number theory. In her spare time, she enjoys playing board games and watching hockey.

Refreshments:

Pizza will be served after the talk.