Speaker
Sudhir Ghorpade
Title
Determinantal hyperplanes over finite fields
Abstract
Just as a hyperplane is given by the zero-set of a linear equation in several variables, a determinantal hyperplane is given by the zero-set of a linear equation in the minors (of various sizes) of a matrix with variable entries. We shall consider the case when the base field is a finite field and discuss the following seemingly natural questions for a fixed generic matrix.
1. What is the maximum possible number of points on a determinantal hyperplane?
2. What are the determinantal hyperplanes where the maximum is attained and what is the number of such maximal determinantal hyperplanes?
3. What can we say about the automorphisms of the ambient space that preserve the determinantal hyperplanes?
Answers to these questions are almost obvious in the case of classical hyperplanes, but appear to be rather nontrivial in the case of determinantal hyperplanes. Connections of these questions with affine open subsets of Grassmann varieties and the theory of error correcting codes will also be indicated.
This is a joint work with Peter Beelen and Tom Hoeholdt.