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: Intersections of Nonclassical Unitals and Conics in PG(2, q 2). | Intersections of Nonclassical Unitals and Conics in PG 2 q2 Angela Aguglia and Vincenzo Giordano Dipartimento di Matematica Politecnico di Bari Via G. Amendola 126 B 70126 Bari Italy Submitted Jun 28 2009 Accepted Sep 1 2010 Published Sep 13 2010 Mathematics Subject Classification E5120 E5121 Abstract In PG 2 q2 with q 2 we determine the possible intersections of a nonclassical Buekenhout-Metz unital U and a conic passing through the point at infinity of U. 1 Introduction The study of the intersection patterns of two relevant geometric objects provides in some cases interesting combinatorial characterizations. An open and rather difficult problem is the classification of all possible intersections between an oval Q and a unital U in a finite projective plane. When Q is a conic and U is a Hermitian or classical unital of PG 2 q2 a complete classification of their intersections is given in 1 . In the present paper we make some advances in this direction by considering a suitable family of nonclassical Buekenhout-Metz unitals in PG 2 q2 with q any prime power. We will use the following representation of a Buekenhout-Metz unital in PG 2 q2 due to Baker and Ebert for q odd and to Ebert for q even. Let z x y denote homogeneous coordinates for points of PG 2 q2 . The line i-. z 0 will be taken as the line at infinity whereas I E will denote the point 0 0 1 . For q 2h let Co be the additive subgroup of GF q defined by Co x E GF q tr x 0 where tr ÍGF q GF 2 lx x x2 . x2 Research supported by the Italian Ministry MIUR Strutture geometriche combinatoria e loro ap-plicazioni. THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R123 1 is the trace map from GF q onto GF 2 . Proposition . Baker and Ebert 3 Ebert 4 5 . Let a b G GF q2 . The point set Ua b 1 t at2 btq 1 r 1t G GF q2 r G GF q u x is a Buekenhout-Metz unital in PG 2 q2 if and only if either q is odd and 4aq 1 bq b 2 is a nonsquare in GF q or q is even b G GF q and aq 1 bq b
