# DIMACS Discrete Math/Theory of Computing Seminar

## Title:

Blocking sets in projective planes

## Speaker:

- Tamas Szonyi
- Budapest and Yale

## Place:

- DIMACS Seminar Room 431, CoRE Building
- Busch Campus, Rutgers University

## Time:

- 4:30 PM
- Tuesday, April 2, 1996

## Abstract:

A blocking set in PG(2,q) (or in a projective plane of order q)
is a set of points intersecting each line. A blocking set is minimal,
when no proper subset of it is a blocking set. The smallest blocking set
is a line. Using purely combinatorial techniques Bruen
(and independently Pelik\'an) proved in 1970 that a blocking set
(not containing a line) in a plane of order q contains at least
q+\sqrt q+1 points. In case of equality the blocking set must
be a subplane of order \sqrt q. Hence Bruen's result is sharp for
PG(2,q), q square. Further combinatorial results
were obtained later by Bruen--Thas, Bruen--Silverman, Bierbrauer and Kitto.
In case of affine planes Jamison and independently Brouwer--Schrijver
proved that a blocking set of AG(2,q) contains at least 2q-1 points.
This is sharp: take a line and one-one point on each line parallel to it.
Their proof used polynomials over GF(q) in an ingenious way. This method
is quite characteristic for Galois geometry. If one studies combinatorial
properties of point sets in a projective plane the proofs often combine
purely combinatorial arguments with algebraic tools using the coordinatizing
field. A classical example is Segre's
theory of complete arcs: to an arc an algebraic curve is associated which
reflects some geometric properties of the arc.

In 1994 Blokhuis improved substantially the previously known bounds if q
is not a square. His result is particularly attractive if q=p is a prime:
a blocking set of PG(2,q) not containing a line contains at least 3(p+1)/2
points and this is sharp.
In Blokhuis' proof lacunary polynomials, introduced by R\'edei,
played a crucial role.

Recently, the present author introduced a method of associating a pair
of curves to a minimal blocking set. The points of both curves
correspond to lines intersecting the blocking set in more than one point.
In particular, the two curves have the same set of GF(q)-rational
points. Using this pair of curves one can prove that a blocking set
of size less than 3(q+1)/2 intersects each line in 1 modulo p points
confirming a conjecture of Blokhuis. In the particular case q=p^2 one can
also show that a minimal blocking set (which is not a line) is either a
subplane of order \sqrt q or contains at least 3(q+1)/2 points.

Document last modified on March 25, 1996