Generalizing Korchmáros-Mazzocca arcs

In this paper, we generalize the so called Korchmáros-Mazzocca arcs, that is, point sets of size $q+t$ intersecting each line in 0, 2 or t points in a finite projective plane of order q. When t is not 2 then this means that each point of the point set is incident with exactly one line meeting the point set in t points. In PG(2,p^n), we change 2 in the definition above to any integer m and describe all examples when m or t is not divisible by p. We also study mod p variants of these objects, give examples and under some conditions we prove the existence of a nucleus