Simple expressions of the Buratti–Del Fra dual hyperoval and the deformation of the Veronesean dual hyperoval

Abstract

Abstract The concept of a dimensional dual hyperoval was introduced by Huybrechts and Pasini in 1999 in [C. Huybrechts and A. Pasini, Frag-transitive extensions of dual affine spaces, Beitrage Algebra Geom. 40 (1999), 503–532]. It is conjectured in [S. Yoshiara, Ambient spaces of dimensional dual arcs, Journal of Algebraic Combinatorics 19 (2004), 5–23] that, if a d-dimensional dual hyperoval exists in P G ( n , 2 ) , then 2 d ⩽ n ⩽ d ( d + 3 ) / 2 . Known d-dimensional dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) are (a) Huybrechtsʼ dual hyperoval [C. Huybrechts, Dimensional dual hyperovals in projective spaces and c.AC* geometries, Discrete Math. 255 (2002), 503–532], (b) Buratti and Del Fraʼs dual hyperoval [M. Buratti and A. Del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003), 245–253], [A. Del Fra and S. Yoshiara, Dimensional dual hyperovals associated with Steiner systems, European J. Combin. 26 (2005), 173–194], (c) the Veronesean dual hyperoval [J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans V n 2 n , The Quarterly Journal of Mathematics, Oxford, 55 (2004), 99–113], [S. Yoshiara, Ambient spaces of dimensional dual arcs, Journal of Algebraic Combinatorics 19 (2004), 5–23], and (d) the deformation of the Veronesean dual hyperoval [H. Taniguchi, A new family of dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) with d ⩾ 3 , Discrete Mathematics 309 (2009), 418–429]. (b) and (d) are originally constructed using complicated expressions, so it has been difficult to deal with these dual hyperovals until now. In this article, we present simple expressions of (b) and (d), which enable us to calculate these dual hyperovals easily.