Linear sets in finite projective spaces

Polverino, Olga

doi:10.1016/j.disc.2009.04.007

Cited by 127 publications

(114 citation statements)

References 69 publications

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“…The same notation and terminology is used when U is a subspace of the vector space V (rt, q) instead of a projective subspace. If moreover we want to mention the dimension of U , we call B(U ) a linear set of rank [8] was one of the first to give importance to these linear sets, and in the last ten years linear sets have played an important role in Finite Geometry, for an overview of applications and connections we refer to [11]. The algebraic connection between linear sets and finite semifields has been successfully used in recent years, see e.g.…”

Section: Semifield Spreads Desarguesian Spreads and Linear Setsmentioning

confidence: 99%

Finite semifields with a large nucleus and higher secant varieties to Segre varieties

Lavrauw

2011

Advg

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Abstract. In [2] a geometric construction was given of a finite semifield from a certain configuration of two subspaces with respect to a Desarguesian spread in a finite-dimensional vector space over a finite field. Moreover, it was proved that any finite semifield can be obtained in this way. In [7] we proved that the configuration needed for the geometric construction given in [2] for finite semifields is equivalent with an (n−1)-dimensional subspace skew to a determinantal hypersurface in PG(n 2 − 1, q), and provided an answer to the isotopism problem in [2]. In this paper we give a generalisation of the BEL-construction using linear sets, and then concentrate on this configuration and the isotopism problem for semifields with nuclei that are larger than its centre. Finite semifieldsA finite semifield S is a finite division algebra, which is not necessarily associative, i.e., an algebra with at least two elements, and two binary operations + and •, satisfying the following axioms: (S1) (S, +) is a group with identity element 0;A finite field is of course a trivial example of a semifield. The first non-trivial examples of semifields were constructed by Dickon in [5]. One easily shows that the additive group of a semifield is elementary abelian, and the additive order of the elements of S is called the characteristic of S. Contained in a semifield are the following important substructures, all of which are isomorphic to a finite field. The left nucleus N l (S), the middle nucleus *

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Section: Semifield Spreads Desarguesian Spreads and Linear Setsmentioning

confidence: 99%

Finite semifields with a large nucleus and higher secant varieties to Segre varieties

Lavrauw

2011

Advg

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“…In other words, Λ may be regarded as the set of all points of PG (r − 1, q t ) whose coordinates are defined by a vector space W over F q of dimension m + 1. Linear sets are used for several remarkable constructions in finite geometry; see [13] for a survey.…”

Section: ])mentioning

confidence: 99%

“…Thus, we can assume that Π contains at least a point P such that the spread element through P intersects Π only in P . According to the terminology of [13], P is a point of the linear set of weight 1. Suppose now that Π is not spanned by its points of weight 1.…”

Section: Is Spanned By Its Totally Decomposable Vectors That Is Itsmentioning

confidence: 99%

On some subvarieties of the Grassmann variety

Giuzzi¹,

Pepe²

2014

Linear and Multilinear Algebra

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Let S be a Desarguesian (t − 1)-spread of PG (rt − 1, q), Π a m-dimensional subspace of PG (rt−1, q) and Λ the linear set consisting of the elements of S with non-empty intersection with Π. It is known that the Plücker embedding of the elements of S is a variety of PG (r t − 1, q), say Vrt. In this paper, we describe the image under the Plücker embedding of the elements of Λ and we show that it is an m-dimensional algebraic variety, projection of a Veronese variety of dimension m and degree t, and it is a suitable linear section of Vrt.

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“…For an overview of the use of linear sets in various other areas of Finite Geometry, we refer to [43], [50], and [62].…”

Section: Semifields: a Geometric Approachmentioning

confidence: 99%