Hermitian Veronesean Caps

Schillewaert, Jeroen; Maldeghem, Hendrik Van

doi:10.1007/978-1-4614-0709-6_9

Cited by 8 publications

(30 citation statements)

References 5 publications

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“…Thas & Van Maldeghem [18] generalized this to the class of ovoids of finite 3-space, whereas Schillewaert & Van Maldeghem [12] proved the same characterization for Hermitian Veronese varieties over any field, as C-Veronesean sets, where C is the class of all ovoids of PG(3, K), with K any skew field.…”

Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 91%

“…Our claim is proved. Now, again by the second main result of [13], G is isomorphic to the quadric Veronese variety of the projective plane PG(3, K). Adding t, we see that N = 6.…”

Section: Lemma 34 Letmentioning

confidence: 94%

“…Let us start by briefly recalling the "classical" Mazzocca-Melone axioms, in their simplest form, namely, for the quadric Veronesean variety X of a projective plane PG(2, K) in the 5-dimensional projective space PG(5, K) (see [13]). One hypothesizes a family Π of planes each member of which intersects X in a conic (or, more generally, an oval, see below).…”

Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 99%

“…This was later generalized by Hirschfeld & Thas to include all finite fields [7]. Thas & Van Maldeghem [16] then considered the case where C is the class of finite plane ovals (and proved each of these ovals must automatically be a conic; an oval is a set of points of a projective plane no three on a line and through each of its point a unique line intersecting the oval in one point), whereas Schillewaert & Van Maldeghem [13] classified C-Veronesean sets for C the class of all ovals of any (finite or infinite) projective plane.…”

Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 99%

“…Consequently, we see that P spans a 5-space, and so it defines a Veronesean embedding of PG(2, K). By [13], P defines a quadric Veronese variety V.…”

Section: Proofmentioning

confidence: 99%

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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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Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 91%

“…Our claim is proved. Now, again by the second main result of [13], G is isomorphic to the quadric Veronese variety of the projective plane PG(3, K). Adding t, we see that N = 6.…”

Section: Lemma 34 Letmentioning

confidence: 94%

Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 99%

Section: Mazzocca-melone Axioms and C-veronesean Setsmentioning

confidence: 99%

“…Consequently, we see that P spans a 5-space, and so it defines a Veronesean embedding of PG(2, K). By [13], P defines a quadric Veronese variety V.…”

Section: Proofmentioning

confidence: 99%

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Projective planes over 2-dimensional quadratic algebras

Schillewaert

Maldeghem

2014

Advances in Mathematics

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Generalized Veronesean embeddings of projective spaces

Thas

Maldeghem

2011

Combinatorica

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We classify all embeddings theta: PG(n, q) -> PG(d, q), with d >= n(n+3)/2 such that theta maps the set of points of each line to a set of coplanar points and such that the image of theta generates PG(d, q). It turns out that d = 1/2n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well

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A characterization of d-uple Veronese varieties

Schillewaert

Struyve

2015

Comptes Rendus. Mathématique

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Presented by Claire VoisinWe characterize d-uple Veronese embeddings of finite-dimensional projective spaces. The easiest non-trivial instance of our theorem is the embedding of the projective plane in a 5-dimensional projective space, a result obtained in 1901 by Severi when the underlying field is the field of complex numbers. © 2015 Published by Elsevier Masson SAS on behalf of Académie des sciences.r é s u m é Nous caractérisons les plongements d-uples de Veronese d'espaces projectifs de dimension finie. L'instance non triviale la plus simple de notre théorème est le plongement du plan projectif dans un espace projectif de dimension 5, un résultat obtenu en 1901 par Severi lorsque le corps sous-jacent est le corps des nombres complexes.

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Hermitian Veronesean Caps

Cited by 8 publications

References 5 publications

Projective planes over 2-dimensional quadratic algebras

Projective planes over 2-dimensional quadratic algebras

Generalized Veronesean embeddings of projective spaces

A characterization of d-uple Veronese varieties

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