Description
A KM-arc of type (0,2,t) is a point set S of a finite projective plane of order q such that each point Q of S is incident with a unique line meeting S in t points and the other lines incident with Q meet S in 2 points. These objects have been studied first by Korchmáros and Mazzocca in 1990, that is why nowadays they are called KM-arcs. KM-arcs exist only for q and t even and they have been studied mostly in Desarguesian planes, where Gács and Weiner proved that the t-secants of a KM-arc are concurrent, their common point is called a nucleus.
In this talk we introduce the following three generalisations of KM-arcs:
1) A generalised KM-arc of type (0,m,t) is a point set S of a finite projective plane of order q such that each point Q of S is incident with a unique line meeting S in t points and the other lines incident with Q meet S in m points.
2) A mod p generalised KM-arc of type (0,m_p,t_p) is a point set S of a finite projective plane of order q=p^n such that each point Q of S is incident with a unique line meeting S in t mod p points and the other lines incident with Q meet S in m mod p points.
3) A mod p generalised KM-arc is a point set S of a finite projective plane of order q=p^n such that for each point Q of S there exists an integer m_Q such that all but at most one of the lines incident with Q meet S in m_Q mod p points.
At the first part of the talk I will present various examples of these objects, some of them are also related to the Dirac-Motzkin conjecture on the minimum number of lines meeting an n-set of the real projective plane in exactly two points. Next I will show some combinatorial results and characterisations. Then, by using Rédei polynomials, I will show that whenever p is not a divisor of m-t, then the m mod p secants have a nucleus, more precisely: let r be a line meeting a mod p generalized KM-arc of type (0,m_p,t_p) in m mod p points, then the t mod p secants meeting S in r are concurrent. This result can be viewed as a generalisation of the fact that an affine (q-1)-set can be extended into an affine q-set such that the two point sets determine the same set of directions (this was proved first by Blokhuis and then generalized for smaller point sets by Szőnyi and by Sziklai).
The fact that the m mod p secants have a nucleus turned out to be very useful in the characterisation of mod p generalised KM-arcs of type (0,m_p,t_p).
Combining this result with results on the stability of k mod p multisets by Szőnyi and Weiner, we were able to describe all generalised KM-arcs of type (0,m,t) when p does not divide m-t.
The proof of this result together with some further characterisations for mod p generalised KM-arcs of type (0,m_p,t_p) will be presented by Zsuzsa Weiner in a future seminar.