Special Sets of the Hermitian Surface and Indicator Sets Antonio Cossidente, 1 Giuseppe Marino, 2 Olga Polverino 3 1 Dipartimento di Matematica, Universit ` a della Basilicata, Contrada Macchia Romana, 85100 Potenza, Italy, E-mail: cossidente@unibas.it 2 Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Universit` a degli Studi di Napoli “Federico II”, Complesso Monte S. Angelo, 80126 Napoli, Italy, E-mail: giuseppe.marino@unina.it 3 Dipartimento di Matematica, Seconda Universit ` a di Napoli, Via Vivaldi, 43, 81100 Caserta, Italy, E-mail: olga.polverino@unina2.it; opolveri@unina.it Received March 14, 2006; revised December 21, 2006 Published online 27 October 2007 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.20148 Abstract: An interesting connection between special sets of the Hermitian surface of PG(3,q 2 ), q odd, (after Shult [13]) and indicator sets of line-spreads of the three-dimensional projective space is provided. Also, the CP-type special sets are characterized. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 18–24, 2008 Keywords: Hermitian surface; special set; ovoid; commuting polarities 1. INTRODUCTION In an article of 1995, E.E. Shult and J.A. Thas [14] give a construction of generalized quadrangles of order (q 2 ,q 2 ) provided the quadric Q - (5,q) admits a collection F of q 2 + 1 pairwise disjoint totally singular lines such that any line of Q - (5,q) not in F meets either 0 or 2 lines of F. Under the Pl¨ ucker map, the set F corresponds to a set S of points of the Hermitian surface H(3,q 2 ) (the dual of Q - (5,q)) with the following properties: (1) |S|= q 2 + 1; (2) any point of H(3,q 2 ) \ S is perpendicular to 0 or 2 points of S, or equivalently, any three points of S generate a non-tangent plane to H(3,q 2 ). A subset of H(3,q 2 ) satisfying properties (1) and (2) is called a special set in [13]. © 2007 Wiley Periodicals, Inc. 18