Blocking semiovals in \(PG\)(2,7) and beyond (Q1867284)

A blocking semi-oval of a projective plane \(P\) is a set \(S\) of points which is a blocking set (every line of \(P\) has a point of and a point off \(S\)) and a semi-oval (every point of \(S\) is incident with exactly one tangent line). The following results are obtained by the authors: 1. In \(PG(2, 7)\) eleven non-equivalent blocking semi-ovals are constructed. 2. There are seven series of blocking semi-ovals constructed. As an example we quote the construction of one of these series: Let \(P = PG(2, q)\), \(q\) odd and \(q \neq 3^e\). Let \(S\) be the set obtained by deleting the points \((0,1,1), (1,0,1),(1,1,0)\) from the triangle defined by the lines \([1,0,0], [0,1,0], [0,0,1]\) and by adding the points \((1,1,-1), (1,-1,-1),(1,-1,1)\). Then \(S\) is a blocking semi-oval. 3. Every blocking semi-oval \(S\) in a projective plane \(P\) can be extended to a blocking semi-oval in a projective plane \(\overline{P}\) containing \(P\) as a subplane.