Sui <i>q</i>-archi completi in piani non desarguesiani di ordine <i>q</i> dispari

Abstract

By a well know theorem of Segre [5] and G. Tallini [7], the q-arcs of the desarguesian plane $PG(2,q)$, are not complete.In [1],[2],[3] it is shown that this theorem cannot be extended to any non-desarguesian plane.In this paper, the following theorem is proved: Let $ω$ be a complete q-arc of a projective plane ? of order q. Denote by $e_j$ the number of those points P of ? for which the number of tangents of $ω$ passing through P is j. Then $e_{\frac{q+1}{2}}≤ 4$ when $q>15$; $e_{\frac{q+h}{2}}≤ 3$ for $h=3,5,7\ldots$.