A characterization of the ovals in symmetric design (Q914686)

Consider a (v,k,\(\lambda\))-design D such that \(v>k\) and let S be an n-set of points of D, where \(n\geq 1\). S is an (n,m)-arc of D if no \(m+1\) points of S lie on a block of D. An S-tangent is a block of D such that \(| B\cap S| =1\). The number of S-tangents containing a point P of D is called the degree of P relative to S. It is well-known that for an (n,2)- arc of a (v,k,\(\lambda\))-design that \(n\leq 1+\frac{k}{\lambda}\) if k- \(\lambda\) is even and \(\lambda\) is a divisor of k or \(v=3\), \(k=2\), and \(\lambda =1\); otherwise \(n\leq 1+\frac{k-1}{\lambda}\). An (n,2)-arc S is said to be an oval of Type I if \(n=1+\frac{k-1}{\lambda}\) (equivalently, each point of S is contained in a unique S-tangent) and to be an oval of Type II if \(n=1+\frac{k}{\lambda}\) (equivalently, there are no S- tangents). In the paper under review the following result is established: Suppose that S is an n-set of points of a (v,k,\(\lambda\))-design D with \(v>k\) and \(n>1\) such that (i) for each P of D, the degree of P relative to S is 0, 1, 2 or k, and (ii) for each point P of S, the degree of P is positive. Then (a) if D possesses no points of degree k, then each point of S lies on at most two S-tangents; furthermore, if there is a point of S which lies on exactly one S-tangent, then S is an oval of Type I and, if each point of S lies on exactly two S-tangents, then D is a projective plane of order two and \(| S| =2,\) (b) if D possesses a unique point P of degree k, then \(\lambda\leq 2\); furthermore, \(S\cup \{P\}\) is an oval of Type II of a projective plane or biplane of even order or of a (3,2,1)-design, and (c) if D possesses at least two points of degree k then \(| S| \geq 3\) and \(\lambda =1\); furthermore, if there are at least three points of degree k, then D is a projective plane of order two and S is the set of points of a line of D and, if there are exactly two points of degree k \((P_ 1\) and \(P_ 2\), say), then S is a \((q+1,3)\)-arc of a projective plane of order \(q\equiv 0(mod 3)\) whose tangent lines form two pencils with centres \(P_ 1\) and \(P_ 2\).