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Quotients of the deformation of Veronesean dual hyperoval in [formula omitted]

Authors
Journal
Discrete Mathematics
0012-365X
Publisher
Elsevier
Publication Date
Volume
312
Issue
3
Identifiers
DOI: 10.1016/j.disc.2011.02.033
Keywords
  • Dual Hyperoval
  • Veronesean
  • Quotient

Abstract

Abstract Let d ≥ 3 . In P G ( d ( d + 3 ) / 2 , 2 ) , there are four known non-isomorphic d -dimensional dual hyperovals by now. These are Huybrechts’ dual hyperoval by Huybrechts (2002) [4], Buratti-Del Fra’s dual hyperoval by Buratti and Del Fra (2003) [1], Del Fra and Yoshiara (2005) [3], Veronesean dual hyperoval by Thas and van Maldeghem (2004) [9], Yoshiara (2004) [12] and the dual hyperoval, which is a deformation of Veronesean dual hyperoval by Taniguchi (2009) [6]. In this paper, using a generator σ of the Galois group G a l ( G F ( 2 d m ) / G F ( 2 ) ) for some m ≥ 3 , we construct a d -dimensional dual hyperoval T σ in P G ( 3 d , 2 ) , which is a quotient of the dual hyperoval of [6]. Moreover, for generators σ , τ ∈ G a l ( G F ( 2 d m ) / G F ( 2 ) ) , if T σ and T τ are isomorphic, then we show that σ = τ or σ = τ − 1 on G F ( 2 d ) . Hence, we see that there are many non-isomorphic quotients in P G ( 3 d , 2 ) for the dual hyperoval of [6] if d is large.

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