Bence Csajbók: Small semiarcs with a long secant in PG(2,q)
A t-semiarc is a pointset S with the property that the number of
tangent lines to S at each of its points is t.
The smallest example for a t-semiarc is a (q+2-t)-arc, but we know
little about the second smallest one.
Actually it is an open question even if t equals one, i.e. in the case
of semiovals.
The known infinite families of small semiarcs, that are not arcs, have
an interesting property:
They have one or two large collinear subsets.
We will show that if the size of a t-semiarc S is small and it has a
large collinear subset, then one of the following holds:
- S has two (q-t)-secants (this type of t-semiarcs have been
completely described previously by Csajbók),
- S is the symmetric difference of a Baer subplane B and one of its
lines l, with t-t points deleted from B\l and from l\B.
In the proof we use the Szőnyi-Weiner Lemma to associate a small
blocking set to the semiarc.
After this we use theorems about small blocking sets.
This is a joint work with Tamás Héger and Gyuri Kiss.