Sets in a finite plane with few intersection numbers and a distinguished point

AbstractThere have been many characterizations of the classical curves in PG(2, q) given by the zeros of quadratic and Hermitian forms. The next step is to characterize pencils of such curves. Here it is done in the case that the pencils have a single base point. A key result that emerges from the investigation is that a certain nonclassical unital is simply the union of conics with a common point