Note on the structure of semiovals in finite projective planes (Q1199581)

A subset \(S\) of the point set of a projective plane \(\Pi\) is called a semioval, if for each point \(s\in S\) there is exactly one line \(T\) (called tangent) such that \(S\cap T=\{s\}\), and then \(S\) is called regular of character \(a\) if \(a\in\mathbb{N}\) and for all lines \(L\) the cardinality \(| L\cap S|\) is in \(\{0,1,a\}\). The authors prove: If \(\Pi\) is a desarguesian projective plane of finite order \(q\) and \(S\) a regular semioval of character \(a\) of \(\Pi\), then: (1) \(a=2\) (i.e. \(S\) is an ovel) or \(a|(q-1)\) and each point \(x\) with \(x\not\in S\) is on 0 or on a tangent. (2) If \(a>2\) then \(S\) is a unital (i.e. \(S\) is the set of absolute points of a hermitian polarity) or \((a-1,q)=1\) and the tangents of collinear points of \(S\) are concurrent. Their conjecture ``\(S\) is an oval or a unital'' remains open.