Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Arcs with large conical subsets. | Arcs with large conical subsets K. Coolsaet H. Sticker Department of Applied Mathematics and Computer Science Ghent University Krijgslaan 281-S9 B-9000 Gent Belgium Submitted Dec 16 2009 Accepted Jul 29 2010 Published Aug 9 2010 Mathematics Subject Classification 51E21 Abstract We classify the arcs in PG 2 q q odd which consist of q 3 2 points of a conic C and two points not on te conic but external to C or q 1 2 points of C and two additional points at least one of which is an internal point of C. We prove that for arcs of the latter type the number of points internal to C can be at most 4 and we give a complete classification of all arcs that attain this bound. Finally we list some computer results on extending arcs of both types with further points. 1 Introduction Consider the Desarguesian projective plane PG 2 q over the finite field of order q with q odd. For k a positive integer define a k-arc to be a set S of points of PG 2 q of size S k such that no three elements of S are collinear. An arc S is called complete if it is not contained in a bigger arc. When q is odd it is well known that an arc can be of size at most k q 1 and that an arc in that case always coincides with the set of points of some conic C and is complete . It is natural to ask what the second biggest size for a complete arc in PG 2 q is. Removing some points from a conic C yields an arc but this arc is obviously not complete. However removing a sufficient number of points at least q 1 2 as will be shown later it may be possible to extend the set thus obtained to an arc by adding a point that does not belong to C. This new arc might not be complete but can be made complete by adding yet more points. This is the kind of arc we will study in this paper. For many values of q arcs of this type are among the largest ones known. Let S be any arc. Then we define a conical subset of S to be any subset T of S of the form T S n C where C is a conic. In this
