Blocking semiovals in PG(2,7) and beyond

Batten (Australas. J. Combin. 22 (2000) 167) stimulated interest in blocking semiovals, sets which are both blocking sets and semiovals. Dover (European J. Combin. 21 (2000) 571) gave a bound on the size of blocking semiovals and presented a family of blocking semiovals of size 3q−4 valid in PG(2,q) for q≥5. posed the question: how many equivalence classes of blocking semiovals are there in PG(2,7) and how many of them are members of infinite families? Comprehensive computer searches by the authors found eleven blocking semiovals, up to projective equivalence. We found that at least seven of these blocking semiovals are contained in infinite families. This paper presents those families. o prove a robust extension theorem: any blocking semioval in a subplane π of a projective plane Π may be extended to a blocking semioval in Π by adding a suitable collection of points.