Hermitian Veroneseans over finite fields

Cooperstein, Bruce N.; Thas, J. A.; Maldeghem, Hendrik Van

doi:10.1515/form.2004.016

Cited by 14 publications

(68 citation statements)

References 6 publications

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“…, y n,n ) with y i,i = x ixi , y i,j = x ixj +x i x j for i < j, and y i,j = rx ixj +rx i x j for i > j. This is projectively independent of the chosen parameter r (see [2]). Clearly, the inverse image with respect to θ of the intersection of H n with a hyperplane of PG(N, q) is a (not necessarily non-singular) Hermitian variety in PG(n, q 2 ).…”

Section: Definitions and Statement Of The Main Resultsmentioning

confidence: 98%

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Some Characterizations of Finite Hermitian Veroneseans

Thas

Maldeghem

2005

Des Codes Crypt

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We characterize the finite Veronesean H n ⊆ PG(n(n + 2), q) of all Hermitian varieties of PG(n, q 2 ) as the unique representation of PG(n, q 2 ) in PG(d, q), d ≥ n(n + 2), where points and lines of PG(n, q 2 ) are represented by points and ovoids of solids, respectively, of PG(d, q), with the only condition that the point set of PG(d, q) corresponding to the point set of PG(n, q 2 ) generates PG(d, q).Using this result for n = 2, we show that H 2 ⊆ PG(8, q) is characterized by the following properties: (1) |H 2 | = q 4 + q 2 + 1; (2) each hyperplane of PG(8, q) meets H 2 in q 2 + 1, q 3 + 1 or q 3 + q 2 + 1 points;(3) each solid of PG(8, q) having at least q + 3 points in common with H 2 shares exactly q 2 + 1 points with it.

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Section: Definitions and Statement Of The Main Resultsmentioning

confidence: 98%

“…Some basic properties of these objects (mostly in the case of index 2) were collected and proved by Lunardon [5] and by Cossidente and Siciliano [1]. Cooperstein, Thas and Van Maldeghem [2] provided a complete Type 3 characterization for H n . No other characterizations are known.…”

Section: Introductionmentioning

confidence: 99%

Some Characterizations of Finite Hermitian Veroneseans

Thas

Maldeghem

2005

Des Codes Crypt

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“…Moreover this implies that the number of cones nnx 6 has to be an integer. This excludes (g x , n x , n) ∈ {(2, 4, 19); (2,5,23); (3,4,20); (5, 5, 26)}.…”

Section: Proofmentioning

confidence: 99%

Projective planes over 2-dimensional quadratic algebras

Schillewaert

Maldeghem

2014

Advances in Mathematics

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The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [5,20,21]. The geometries appearing in the second row are Severi varieties [24]. We provide an easy uniform axiomatization of these geometries and related ones, over an arbitrary field. In particular we investigate the entry A 2 × A 2 in the magic square, characterizing Hermitian Veronese varieties, Segre varieties and embeddings of Hjelmslev planes of level 2 over the dual numbers. In fact this amounts to a common characterization of "projective planes over quadratic 2-dimensional algebras", in casu the split and non-split Galois extensions, the inseparable extensions of degree 2 in characteristic 2 and the dual numbers.

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“…This section is taken from Cooperstein, Thas and Van Maldeghem [1]; see also Cossidente and Siciliano [2] for the case n = 2. We do not claim that we are the first to study Hermitian Veroneseans nor that (part of) Section 6.2 cannot be found elsewhere.…”

Section: Introduction To Hermitian Veroneseans Over Finite Fieldsmentioning

confidence: 99%

“…In Cooperstein, Thas and Van Maldeghem [1] it is shown that a Hermitian set is a cap and consequently in what follows a Hermitian set is referred to as a Hermitian cap. The reader should compare this with the definition of a Veronesean cap in Section 2.…”

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Characterizations of quadric and Hermitian Veroneseans over finite fields

Thas¹,

Maldeghem²

2003

J.Geom.

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In Hirschfeld and Thas [5] the most important characterizations of quadric Veroneseans are surveyed. However a few difficult cases were still open, in particular the even case. In [10,11] Thas and Van Maldeghem not only solve all open cases, but they also generalize most of these characterizations in several ways: they do not restrict themselves to the quadric Veronesean of the plane PG(2, q), they allow ovals instead of conics, and they also characterize projections of quadric Veroneseans. Further, Cooperstein, Thas and Van Maldeghem [1] contains some properties of Hermitian Veroneseans over finite fields and also these varieties and some of their projections are characterized. All these results on Veroneseans will be surveyed here.

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Hermitian Veroneseans over finite fields

Cited by 14 publications

References 6 publications

Some Characterizations of Finite Hermitian Veroneseans

Some Characterizations of Finite Hermitian Veroneseans

Projective planes over 2-dimensional quadratic algebras

Characterizations of quadric and Hermitian Veroneseans over finite fields

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