Mathematics Graduate Student Colloquium

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

Counting n-arcs in projective planes

Kelly Isham
Thursday, November 7, 2019
12:30 pm - 12:50 pm
RH 340N

Talk Abstract:

An n-arc in \PP^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. He found that for n less than or equal to 6, the formula for the number of n-arcs is a polynomial in q and that for n=7,8, the formula is quasi-polynomial in q. In 1995, Iampolskaia et al showed that the formula for n=9 is also quasipolynomial. Recent work by Kaplan et al extended this result to arbitrary projective planes of order q. This leads to the question - will the number of n-arcs over \PP^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 n-arcs in the projective plane over F_q is equivalent to counting the number of rational points on certain varieties over F_q. We use this new approach to prove that the number of 10-arcs in a projective plane over F_q is not quasipolynomial. Lastly, 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 4th year Ph.D candidate interested in number theory. In her spare time, she enjoys playing board games and watching hockey.

Refreshments:

Pizza will be served after the talk.