Primitive arcs in \(\mathrm{PG}(2,q)\)
A \(k\)-arc \(K\) in a finite projective plane is a set of \(k\) points, no three of which are collinear. A \(k\)-arc is said to be complete if it cannot be included in a \(k+1\) arc of the same plane. This paper gives the classification of all complete \(k\)-arcs in \(\mathrm{PG}(2,q)\) which are invariant under a transitive projective group active primitively on the points of the arc. Let \(G\) be the maximum subgroup \(\mathrm{PGL}(3,q)\) which leaves \(K\) invariant. There are two main cases: I. \(G\) contains an elementary abelian group which acts regularly on \(K\). II. \(G\) is an almost simple group with socle \(S\), where \(S\) is nonabelian simple; \(G \subseteq \Aut S\). There are three types of possibilities for \(S\), namely \(L_3(q')\), \(U_3(q')\) and \(L_2(q')\) for suitable \(q'\) dividing \(q\). Most of the arcs are conics in \(\mathrm{PG}(2,q)\), conics in some subplane, hyperovals, orbits under a Singer group or are sporadic cases with relatively small \(k\).
scientific article; zbMATH DE number 734533
scientific article