Ilaria Cardinali

Semifield planes of order q 4 with kernel F q 2 and center F q

A classification of semifield planes of order q4 with kernel Fq2 and center Fq is given. For q an odd prime, this proves the conjecture stated in [M. Cordero, R. Figueroa, On the semifield planes of order 54 and dimension 2 over the kernel, Note Mat. (in press)]. Also, we extend the classification of semifield planes lifted from Desarguesian planes of order q2, q odd, obtained in [M. Cordero, R. Figueroa, On some new classes of semifield planes, Osaka J. Math. 30 (1993) 171-178], to the even characteristic case.

The structure of full polarized embeddings of symplectic and Hermitian dual polar spaces

Abstract Let Δ be a thick dual polar space of rank n ≥ 2 and let e be a full polarized embedding of Δ into a projective space ∑. For every point x of Δ and every i ∈ {0, …, n}, let T i (x) denote the subspace of ∑ generated by all points e(y) with d(x, y) ≤ i. We show that T i (x) does not contain points e(z) with d(x, z) ≥ i + 1. We also show that there exists a well-defined map ei x from the set of (i – 1)-dimensional subspaces of the residue Res Δ(x) of Δ at the point x (which is a projective space of dimension n – 1) to the set of points of the quotient space T i (x)/T i –1 (x). In this paper we study the structure of the maps ei x and the subspaces Ti(x) for some particular full polarized embeddings of the symplectic and the Hermitian dual polar spaces. Our investigations allow us to answer some questions asked in the literature.

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose @D is a dual polar space of rank n and H is a hyperplane of @D. Cardinali, De Bruyn and Pasini have already shown that if n>=4 and the line size is greater than or equal to 4 then the hyperplane complement @D-H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.

Spreads in H(q) and 1-systems of Q(6,q )

In this paper we prove that the projections along reguli of a translation spread of the classical generalized hexagon H(q) are translation ovoids of Q(4, q). As translation ovoids of Q(4, 2r) are elliptic quadrics, this forces that all translation spreads ofH (2r) are semi-classical. By representing H(q) as a coset geometry, we obtain a characterization of a translation spread in terms of a set of points of PG(3, q) which belong to imaginary chords of a twisted cubic and we construct a new example of a semi-classical spread ofH (2r). Finally, we study the semi-classical locally Hermitian 1-systems ofQ (6, q) which are spreads of Q?(5, q).

Codes and caps from orthogonal Grassmannians

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding e k gr of an orthogonal Grassmannian Δ k . In particular, we determine some of the parameters of the codes arising from the projective system determined by e k gr ( Δ k ) . We also study special sets of points of Δ k which are met by any line of Δ k in at most 2 points and we show that their image under the Grassmann embedding e k gr is a projective cap.

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n, K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8, K) and show that every locally singular hyperplane of DQ(8, K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.

Veronesean embeddings of dual polar spaces of orthogonal type

Given a point-line geometry Γ and a pappian projective space S, a veronesean embedding of Γ in S is an injective map e from the point-set of Γ to the set of points of S mapping the lines of Γ onto non-singular conics of S and such that e(Γ) spans S. In this paper we study veronesean embeddings of the dual polar space Δn associated to a non-singular quadratic form q of Witt index n⩾2 in V=V(2n+1,F). Three such embeddings are considered, namely the Grassmann embedding engr which maps a maximal singular subspace 〈v1,…,vn〉 of V (namely a point of Δn) to the point 〈⋀i=1nvi〉 of PG(⋀nV), the composition envs:=ν2n∘enspin of the spin (projective) embedding enspin of Δn in PG(2n−1,F) with the quadric veronesean map ν2n:V(2n,F)→V((2n+12),F), and a third embedding e˜n defined algebraically in the Weyl module V(2λn), where λn is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that e˜n and envs are isomorphic. If char(F)≠2 then V(2λn) is irreducible and e˜n is isomorphic to engr while if char(F)=2 then engr is a proper quotient of e˜n. In this paper we shall study some of these submodules. Finally we turn to universality, focusing on the case of n=2. We prove that if F is a finite field of odd order q>3 then e2sv is relatively universal. On the contrary, if char(F)=2 then e2vs is not universal. We also prove that if F is a perfect field of characteristic 2 then envs is not universal, for any n⩾2.

Embeddings of Line-Grassmannians of Polar Spaces in Grassmann Varieties

An embedding of a point-line geometry \(\varGamma \) is usually defined as an injective mapping \(\varepsilon \) from the point-set of \(\varGamma \) to the set of points of a projective space such that \(\varepsilon (l)\) is a projective line for every line \(l\) of \(\varGamma \). However, different situations are considered in the literature, where \(\varepsilon (l)\) is allowed to be a subline of a projective line or a curve. In this paper we propose a more general definition of embedding which includes all the above situations and we focus on a class of embeddings, which we call Grassmann embeddings, where the points of \(\varGamma \) are firstly associated to lines of a projective geometry \({\mathrm {PG}}(V)\), next they are mapped onto points of \({\mathrm {PG}}(V\wedge V)\) via the usual projective embedding of the line-grassmannian of \({\mathrm {PG}}(V)\) in \({\mathrm {PG}}(V\wedge V)\). In the central part of our paper we study sets of points of \({\mathrm {PG}}(V\wedge V)\) corresponding to lines of \({\mathrm {PG}}(V)\) totally singular for a given alternating, hermitian or quadratic form of \(V\). Finally, we apply the results obtained in that part to the investigation of Grassmann embeddings of several generalized quadrangles.

Enumerative coding for line polar Grassmannians with applications to codes

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Minimum distance of Orthogonal Line-Grassmann Codes in even characteristic

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