Algebraický seminář

Suhir Ghorpade (IIT Bombay): Maximal projective algebraic varieties over finite fields, Hilbert functions, and applications

Abstract: Let F be a finite field and let X be a projective algebraic variety defined over F. When X is irreducible, or better still, nonsingular, there are good estimates for the number of F-rational points of X using deeper methods in algebraic geometry. But what if X is reducible? More specifically, we are interested in the following question: Let m, d and r be positive integers. What is the maximum number of F-rational points on an algebraic variety (in the m-dimensional projective space) given by the common zeros of r linearly independent homogeneous polynomials of degree d in m + 1 variables with coefficients in F?
This question was first raised by M. Tsfasman in the case of a hypersurface, i.e., when r = 1. This case was soon settled by J.-P. Serre (1991). Later Tsfasman together with M. Boguslavsky formulated a remarkable conjecture in the general case, and this was shown to hold in the affirmative in the next case of r = 2 by Boguslavsky (1997). Then about two decades later, it was shown that the conjecture is valid if the number of polynomials is at most the number of variables, i.e., r ≤ m + 1, but the conjecture can be false in general. Newer conjectures were then formulated and although there has been considerable progress concerning them, the general case is still open. Some of the newer conjectures and results use Hilbert functions and techniques from extremal combinatorics. The above question is also intimately related to the study of maximal sections of Veronese varieties by linear subvarieties of the ambient projective space. Moreover, answers to this question can be applied to the study of an important class of linear error correcting codes, called projective Reed-Muller codes.