Innovations in Incidence Geometry

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

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
First available in Project Euclid: 28 February 2019

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

References

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