Innovations in Incidence Geometry

New quotients of the $d$-dimensional Veronesean dual hyperoval in $\mathrm{PG}(2d+1,2)$

Hiroaki Taniguchi and Satoshi Yoshiara

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Abstract

Let d 3 . For each e 1 , Thas and Van Maldeghem constructed a d -dimensional dual hyperoval in PG ( d ( d + 3 ) 2 , q ) with q = 2 e , called the Veronesean dual hyperoval. A quotient of the Veronesean dual hyperoval with ambient space PG ( 2 d + 1 , q ) , denoted S σ , is constructed by Taniguchi, using a generator σ of the Galois group Gal ( GF ( q d + 1 ) GF ( q ) ) .

In this note, using the above generator σ for q = 2 and a d -dimensional vector subspace H of GF ( 2 d + 1 ) over GF ( 2 ) , we construct a quotient S σ , H of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) in case d is even. Moreover, we prove the following: for generators σ and τ of the Galois group Gal ( GF ( 2 d + 1 ) GF ( 2 ) ) ,

  1. S σ above (for q = 2 ) is not isomorphic to S τ , H ,
  2. S σ , H is isomorphic to S σ , H for any d -dimensional vector subspaces H and H of GF ( 2 d + 1 ) , and
  3. S σ , H is isomorphic to S τ , H if and only if σ = τ or σ = τ 1 .

Hence, we construct many new non-isomorphic quotients of the Veronesean dual hyperoval in PG ( 2 d + 1 , 2 ) .

Article information

Source
Innov. Incidence Geom., Volume 12, Number 1 (2011), 151-165.

Dates
Received: 5 January 2011
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323073

Digital Object Identifier
doi:10.2140/iig.2011.12.151

Mathematical Reviews number (MathSciNet)
MR2942722

Zentralblatt MATH identifier
1293.51003

Subjects
Primary: 05BXX 05EXX 51EXX

Keywords
dual hyperoval Veronesean quotient

Citation

Taniguchi, Hiroaki; Yoshiara, Satoshi. New quotients of the $d$-dimensional Veronesean dual hyperoval in $\mathrm{PG}(2d+1,2)$. Innov. Incidence Geom. 12 (2011), no. 1, 151--165. doi:10.2140/iig.2011.12.151. https://projecteuclid.org/euclid.iig/1551323073


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References

  • A. Del Fra, On $d$-Dimensional Dual Hyperovals, Geometriae Dedicata, 79 (2000), 157–178.
  • C. Huybrechts and A. Pasini, Flag transitive extensions of dual affine spaces, Contributions to Algebra and Geometry, 40 (1999), 503–532.
  • H. Taniguchi, On a family of dual hyperovals over $GF(q)$ with $q$ even, European Journal of Combinatorics, 26 (2005), 195–199.
  • H. Taniguchi, On isomorphism problem of some dual hyperovals in PG(2d+1,q) with q even, Graphs and Combinatorics, 23 (2007), 455–465.
  • J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans $\mathcal{V}_n^{2^n}$, The Quarterly Journal of Mathematics, Oxford, 55 (2004), 99–113.
  • S. Yoshiara, A family of $d$-dimensional dual hyperovals in $PG(2d+1,2)$, European Journal of Combinatorics, 20 (1999), 589–603.
  • S. Yoshiara, Notes on Taniguchi's dimensional dual hyperovals, European Journal of Combinatorics, 28 (2007), 674–684.