## About

196

Publications

8,615

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Introduction

I see that a few friends are wondering what I am presently doing, which are my research projects, if any. I confess that I have no research project. I have retired in October 2014. As long as I have been on service, I could only work. I never had time to think about what I was doing. Now, I want to exploit the time I still have to understand what I have been doing for so many years. Having been a mathematician, I want to really understand mathematics. And more, of course.

Additional affiliations

November 2018 - present

**no institution**

Position

- Retired

October 2014 - November 2018

**University of Siena**

Position

- Professor Emeritus

November 1993 - September 2014

Position

- Professor (Full)

Description

- Lecturing in Geometry and Linear Algebra, Analysis (occasionally when necessary), Combinatorial Geometry, Number Theory and Cryptography, Graph Theory, Topology. Research on Diagram Geometry and related groups, buildings and polar spaces. Occasionally, joint research on cryptography and applied graph theory.

Education

October 1966 - July 1970

## Publications

Publications (196)

In this paper we investigate hyperplanes of the point-line geometry A_{n,{1,n}}(F) of point-hyerplane flags of the projective geometry PG(n,F). Renouncing a complete classification, which is not yet within our reach, we describe the hyperplanes which arise from the natural embedding of A_{n,{1,n}}(F), that is the embedding which yields the adjoint...

In this paper we consider a family of projective embeddings of the geometry A_{n,{1,n}}(F) of point-hyperplanes flags of PG(n,F). The natural embedding e_1 is one of them. It maps every point-hyperplane flag (p,H) onto the vector-line < x\otimes\xi > where x is a representative vector of p and \xi is a linear functional describing H. The other embe...

In this paper we propose a definition of regularity suited for polar spaces of infinite rank and we investigate to which extent properties of regular polar spaces of finite rank can be generalized to polar spaces of infinite rank.

In this paper we consider a family of projective embeddings of the geometry $\Gamma = A_{n,\{1,n\}}(F)$ of point-hyperplanes flags of the projective geometry $\Sigma = PG(n,F)$. The natural embedding $\varepsilon_{mathrm{nat}}$ is one of them. It maps every point-hyperplane flag $(p,H)$ of $\Sigma$ onto the vector-line $\langle x\otimes\xi\rangle$,...

In this paper we investigate hyperplanes of the point-line geometry $\mathit{A}_{n,\{1,n\}}(\mathbb{F})$ of point-hyerplane flags of the projective geometry $\mathrm{PG}(n,\mathbb{F})$. Renouncing a complete classification, which is not yet within our reach, we describe the hyperplanes which arise from the natural embedding of $\mathit{A}_{n,\{1,n\...

Apolar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG( V ) such that the image ε (𝒮) of 𝒮 by ε is defined by an alternating form of V . In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially whe...

Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K. As a consequence, the generating rank of Γ(K) is...

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especia...

In this paper we compute the generating rank of k -polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k -Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2 n . We also study generating sets for the 2-Grassmannians arising from qu...

Let Γ be an embeddable non-degenerate polar space of finite rank n≥2. Assuming that Γ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least 5 and certain generalized quadrangles defined over quaternion division rings), let ε:Γ→PG(V) be the universal embedding of Γ. Let S be a subspace of Γ and...

We consider various regular graphs defined on the set of elements of given rank of a finite polar space. It is likely that no two such graphs, of the same kind but defined for different ranks, can have the same degree. We shall prove this conjecture under the hypothesis that the considered rank are not too small.

Let $\Gamma$ be an embeddable non-degenerate polar space of finite rank $n \geq 2$. Assuming that $\Gamma$ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least $5$ and certain generalized quadrangles defined over quaternion division rings), let $\varepsilon:\Gamma\to\mathrm{PG}(V)$ be the univ...

The rank of a point-line geometry \(\Gamma \) is usually defined as the generating rank of \(\Gamma \), namely the minimal cardinality of a generating set. However, when the subspace lattice of \(\Gamma \) satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of \(\Gamma \) and take the least upper...

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic 2 case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a...

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given division ring $K$ (a field when $X_n = D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $\Gamma(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$...

The rank of a point-line geometry Γ is usually defined as the generating rank of Γ, namely the minimal cardinality of a generating set. However, when the subspace lattice of Γ satisfies the Exchange Property we can also try a different definition: consider all chains of subspaces of Γ and take the least upper bound of their lengths as the rank of Γ...

In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising f...

This paper is a revised version of a paper with the same title already inserted in my RG list of contributions. Here is the abstract: a classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group acting continuously on the geometry, has been ob...

This is a revised version of a paper made available at http://arxiv.org/abs/1811.04832. A slightly extended version of it will soon appear in IIG. Here is a description of the paper. A classification of homogeneous compact Tits geometries of irreducible spherical type, with connected panels and admitting a compact flag-transitive automorphism group...

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic $2$ case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define...

Consider the Grassmann graph formed by $k$-dimensional subspaces of an $n$-dimensional vector space
over the field of $q$ elements ($1<k<n-1$) and
denote by $\Pi(n,k)_q$ the restriction of this graph to the set of projective $[n,k]_q$ codes.
In the case when $q\ge \binom{n}{2}$, we show that
the graph $\Pi(n,k)_q$ is connected, its diameter is eq...

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $\varphi$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let $\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)$ be the Plucker embedding of the $k$-Grassman...

The aim of this volume is to help the reader to realize that seldom we can give problems precise solutions, even when we believe to hold one of them. We must be happy with approximate solutions. Various methods to accomplish the latter are examined in this book and compared. The book is written in italian. The uploaded text is a preliminary version...

We introduce the class of transparent embeddings for a point-line geometry $\Gamma = ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $\Gamma$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $\Gamma$. We will then investigate the transparency of Pl\"uc...

This is a preliminary sketchy draft of a paper later published in Linear and Multilinear Algebra. The main result of the paper is the following: given a vector space $V$ defined over a field F and a sesquilinear form f on V, a tensor equation is obtained that caracterizes the set of $k$-subspaces of V which either are totally isotropic for f or mee...

In this paper a tensor equation is given that characterizes the k-grassmannian of a classical polar space as a subvariety of a projective k-grassmannian.

Polar Grassmann codes of orthogonal type have been introduced in [1]. They are sub-codes of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for q odd. [Warni...

Given an $N$-dimensional vector space $V$ over a field $\mathbb{F}$ and a
trace-valued $(\sigma,\varepsilon)$-sesquilinear form $f:V\times V\rightarrow
\mathbb{F}$, with $\varepsilon = \pm 1$ and $\sigma^2 =
\mathrm{id}_{\mathbb{F}}$, let ${\cal S}$ be the polar space of totally
$f$-isotropic subspaces of $V$ and let $n$ be the rank of ${\cal S}$....

For k =1, 2,...., n-1 let V-k = V (lambda(k)) be the Weyl module for the special orthogonal group G = SO(2n + 1, F) with respect to the k-th fundamental dominant weight lambda(k) of the root system of type B-n and put V-n = V(2 lambda(n).) It is well known that all of these modules are irreducible when char (F) not equal 2 while when char. (F) = 2...

An embedding of a point-line geometry $\Gamma$ is usually defined as an injective mapping $\varepsilon$ from the
point-set of $\Gamma$ to the set of points of a projective space such that $\varepsilon(l)$ is a projective line
for every line $l$ of $\Gamma$. However, different situations are considered in the literature, where
$\varepsilon(l)$ is al...

This file contains the preliminary version of a paper published in a special volume of Innovations in Incidence Geometry ("50 years of finite geometry", vol. 17, 1917, pp.31-72).
In this paper we introduce generalized pseudo-quadratic forms and develope
some theory for them. Recall that the codomain of a
$(\sigma,\varepsilon)$-quadratic form is th...

Given a non-singular quadratic form q of maximal Witt index on
V := V(2n+1, F), let Δ be the building of type Bn
formed by the subspaces of V totally singular for q and, for 1≤k≤n, let Δk
be the k-grassmannian of Δ. Let εk be the embedding of Δk
into PG(⋀kV) mapping every point 〈v1,v2,…,vk〉 of Δk
to the point 〈v1∧v2∧⋯∧vk〉 of PG(⋀kV). It is known t...

The classification of finite flag-transitive linear spaces, obtained by Buekenhout, Delandtsheer, Doyen, Kleidman, Liebeck and Saxl [20] at the end of the eighties, gave new impulse to the program of classifying various classes of locally finite flag-transitive geometries belonging to diagrams obtained from a Coxeter diagram by putting a label L or...

Given a point-line geometry P and a pappian projective space S,a
veronesean embedding of P in S is an injective map e from the point-set
of P to the set of points of S mapping the lines of P onto non-singular
conics of S and such that e(P) spans S. In this paper we study
veronesean embeddings of the dual polar space \Delta_n associated to a
non-sin...

If a geometry $\Gamma$ is isomorphic to the residue of a point $A$ of a shadow geometry of a spherical building $\Delta$, a representation $\varepsilon_\Delta^A$ of $\Gamma$ can be given in the unipotent radical $U_{A^*}$ of the stabilizer in $\mathrm{Aut}(\Delta)$ of a flag $A^*$ of $\Delta$ opposite to $A$, every element of $\Gamma$ being mapped...

Let V be a 2n-dimensional vector space over a field Fand ξ a non-degenerate alternating form defined on V. Let Δ be the building of type Cn formed by the totally ξ-isotropic subspaces of V and, for 1 ≤ k ≤ n, let Gk and Δk be the k-grassmannians of PG(V) and Δ, embedded in W_k=\wedge^kV
and in a subspace V_k\subseteq W_k respectively, where dim(Vk)...

For $k = 1, 2,...,n-1$ let $V_k = V(\lambda_k)$ be the Weyl module for the
special orthogonal group $G = \mathrm{SO}(2n+1,\F)$ with respect to the $k$-th
fundamental dominant weight $\lambda_k$ of the root system of type $B_n$ and
put $V_n = V(2\lambda_n)$. It is well known that all of these modules are
irreducible when $\mathrm{char}(\F) \neq 2$ w...

An embedding of a point-line geometry \Gamma is usually defined as an injective mapping \epsilon from the point-set of \Gamma to the set of points of a projective space such that \epsilon(l) is a projective line for every line l of \Gamma, but different situations have lately been considered in the literature, where \epsilon(l) is allowed to be a s...

In this paper I survey a number of recent results on projective and Veronesean embeddings of orthogonal Grssmannians and propose a few conjectures and problems.

This paper is a survey of the work done by a number of authors on the classification of flag-transitive geometries with diagram and orders as below.

Let V be a 2n-dimensional vector space over a field F and G the symplectic group Sp(2n,F) stabilizing a non-degenerate alternating form f of V and, for k = 1,..., n let Sk the k-grassmannian of the polar space associated to G, let ek be the natural embedding of Sk in the k-th exterior power of V and W(k) the subspace spanned by the image ek(Sk) in...

Let V be a 2n-dimensional vector space over a field F equipped with a non-degenerate alternating form ξ. Let Gn be the n-grassmannian of PG(V) and Δn the dual of the polar space Δ associated to ξ. Then Gn and Δn are naturally embedded in the vector space Wn = \wedge^nV and Vn\subseteq W_n respectively, where dim(Wn)=\binom{2n}{n} and dim(Vn) = \bin...

Let $V_k$ be the Weyl module of dimension ${2n\choose k}-{2n\choose k-2}$ for the group $G = \mathrm{Sp}(2n,\mathbb{F})$ arising from the $k$-th fundamental weight of the Lie algebra of $G$. Thus, $V_k$ affords the grassmann embedding of the $k$-th symplectic polar grassmannian of the building associated to $G$. When $\mathrm{char}(\mathbb{F}) = p...

A rich information can be found in the literature on Weyl modules for Sp(2n, F), but the most important contributions to this topic mainly enlighten the algebraic side of the matter. In this paper we try a more geometric approach. In particular, our approach enables us to obtain a sufficient condition for a module as above to be uniserial and a geo...

. Many properties of polar spaces of finite rank fail to hold in polar spaces of infinite rank. For instance, in a polar space
of infinite rank it can happen that maximal singular subspaces have different dimensions; every polar space of infinite rank
contains singular subspaces that cannot be obtained as intersections of any family of maximal sing...

Let Γ be the dual of a classical polar space and let e be a projective embedding of Γ, defined over a commutative division ring. We shall prove that, if e is homogeneous, then it is polarized.

Let F be a field of characteristic 2. It is well known that if F is a perfect then the polar space associated to Sp(2n,F) is isomorfic to the polar space associated to SO(2n+1,F). This is no more true when F is non-perfect. We discuss this situation. We show that in this case a quadratic form f still exists defined over an F-vector space and such t...

Let $\Delta$ be a dual polar space of rank \geq 4$, $ be a hyperplane of $\Delta$
and $\Gamma: = \Delta\setminus H$ be the complement of $ in $\Delta$. We shall prove that, if all lines of $\Delta$ have more than $ points, then $\Gamma$ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplane...

Convergence properties of distributed consensus protocols on networks of dynamical agents have been analyzed by combinations of algebraic graph theory and control theory tools under certain assumptions, such as strong connectivity. Strong connectivity can be regarded as the requirement that the information of each agent propagates to all the others...

Let Γ be an extended tilde geometry of rank n>2 such that there exists a 1-covering γ:Γ→Φ where Φ is a c.Cn−1-geometry with orders 1,2,…,2. Suppose that the normalizer in Aut(Γ) of the deck group of γ acts flag-transitively on Γ. We prove that, under these hypotheses, only three possibilities exist for the universal cover of Γ.

We prove that the geometry of vertices, edges and q n -cliques of the graph Alt(n+1,q) of (n+1)-dimensional alternating forms over GF(q), n≥4, is the unique flag-transitive geometry of rank 3 where planes are isomorphic to the point-line system of AG(n,q) and the star of a point is dually isomorphic to a projective space.

Let V be a vector space over a division ring K. Let P be a spanning set of points in Σ:=PG(V). Denote by K(P) the family of sub-division rings F of K having the property that there exists a basis BF of V such that all points of P are represented as F-linear combinations of BF. We prove that when K is commutative, then K(P) admits a least element. W...

We study geometries that arise from the natural G2(K) action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type C3(K) where K is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-...

We introduce the notion of scattered sets of points of a dual polar space, focusing on minimal ones. We prove that a dual polar space Δ of rank n always admits minimal scattered sets of size 2n. We also prove that the size of a minimal scattered set is a lower bound for dim(V) if the dual polar space Δ has a polarized embedding e:Δ→PG(V), namely a...

We prove that every flag-transitive locally finite (PG
*.PG)-geometry is a truncated projective geometry.

We study geometries that arise from the natural $G_2(K)$ action on the
geometry of one-dimensional subspaces, of nonsingular two-dimensional
subspaces, and of nonsingular three-dimensional subspaces of the building
geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One
of these geometries is intransitive in such a way that...

Cooperstein (6), (7) proved that every finite symplectic dual polar space DW (2n-1, q), q > 2, can be generated by B(2n,n)-B(2n,n-2) points, where B(m,k) stands for the binomial of m over k, and that every finite Hermitian dual polar space DH(2n-1, q**2), q> 2, can be generated byB(2n,n) points. In the present paper, we show that these conclusions...

In this paper, a family of nonlinear congruential generators (NLCGs) based on the digitized Reacutenyi map is considered for the definition of hardware-efficient pseudorandom number generators (PRNGs), and a theoretical framework for their study is presented. The authors investigate how the nonlinear structure of these systems eliminates some of th...

This paper is a shortened exposition of the theory of shrinkings, with particular emphasis on the relations between shrinkings and geometries at infinity or affine expansions. To make things easier, we shall only consider locally affine geometries, referring the reader to [A. Pasini, C. Wiedorn, Local parallelisms, shrinkings and geometries at infi...

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphis...

Given a field KK of characteristic 2, let W(2n−1,K)W(2n−1,K) be the symplectic polar space defined in PG(2n−1,K)PG(2n−1,K) by a non-degenerate alternating form of V(2n,K)V(2n,K) and Q(2n,K)Q(2n,K) be the quadric of PG(2n,K)PG(2n,K) associated to a non-singular quadratic form of Witt index n. In the literature, it is often claimed that W(2n−1,K)≅Q(2...

Let Δ be a dual polar space of rank n⩾4n⩾4, H be a hyperplane of Δ and Γ:=Δ\HΓ:=Δ\H be the complement of H in Δ. We shall prove that, if all lines of Δ have more than 3 points, then Γ is simply connected. Then we show how this theorem can be exploited to prove that certain families of hyperplanes of dual polar spaces, or all hyperplanes of certain...

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 h...

A d-dimensional dual hyperoval can be regarded as the image S = ρ(Σ) of a full d-dimensional projective embedding ρ of a dual circular space Σ. The affine expansion Exp(ρ) of ρ is a semibiplane and its universal cover is the expansion of the abstract hull ρ of ρ. In this paper we consider Huybrechts's dual hyperoval, namely ρ(Σ) where Σ is the dual...

It is known that every lax projective embedding $e:\Gamma\rightarrow PG(V)$ of a point-line geometry $\Gamma$ admits a {\em hull}, namely a projective embedding $\tilde{e}:\Gamma\rightarrow PG(\tilde{V})$ uniquely determined up to isomorphisms by the following property: $V$ and $\tilde{V}$ are defined over the same skewfield, say $K$, there is morp...

Given a locally a affine geometry Γ of order 2 and a flag-transitive subgroup G≤Aut(Γ), suppose that the shrinkings of Γ are isomorphic to the a affine expansion of the upper residue of a line of Γ by a homogeneous representation in a 2-group. We shall prove that, under certain hypotheses on the stabilizers G p and G l of a point p and a line l, we...

The Steiner system ∑ = S(12,6,5) admits a unique lax projective embedding / in PG(V), V = F(6,3). The embedding / induces a full projective embedding e of the dual Δ of ∑ in the dual PG(V*) of PG(V). The affine expansion Afe(Δ) of Δ to AG(V*) (also called linear representation of Δ in AG(V*)) is a flag-transitive geometry with diagram and orders as...

Let Γ be a non-degenerate polar space of rank n ≥ 3 where all of its lines have at least three points. We prove that, if Γ admits a lax embedding e : Γ → Σ in a projective space Σ defined over a skewfield K, then Γ is a classical and defined over a sub-skewfield K
0 of K. Accordingly, Γ admits a full embedding e
0 in a K
0-projective space Σ0. We a...

The classification of finite flag-transitive linear spaces, obtained by F. Buekenhout, A. Delandtsheer, J. Doyen, P. B. Kleidman, M. W. Liebeck, and J. Saxl [Geom. Dedicata 36, No. 1, 89–94 (1990; Zbl 0707.51017)] at the end of the eighties, gave new impulse to the program of classifying various classes of locally finite flag-transitive geometries...

An ovoid of a dual polar space is a set of points meeting every line of in exactly one point. In this paper, we consider the dual DW(5,q) of the polar space W(5,q) associated to a non-degenerate alternating form of V(6,q), proving that no ovoids exist in DW(5,q).

In this paper we sketch a general theory of embeddings for geometries with string diagrams, focusing on their hulls. An affine-like geometry, which we call expansion, is associated to every embedding. As we shall prove, the universal cover of the expansion of an embedding is the expansion of the hull of that embedding. Some applications of this the...

We consider extensions of the F4(2)-building with the diagramsuch that the residue of every element of the rightmost type is a one-point extension of the corresponding C3(2)-residue in the building. Four flag-transitive such geometries are known with the automorphism groups isomorphic to 2E6(2):2, 3·2E6(2):2, E6(2):2 and 226:F4(2). The first exampl...

It is well known that, given a point-line geometry and a projective embedding " : ! PG(V ), if dim(V ) equals the size of a generating set of , then " is not derived from any other embedding. Thus, if admits an absolutely universal embedding, then " is absolutely universal. In this paper, without assuming the existence of any absolutely universal e...

The geometries studied in this paper are obtained from buildings of spherical type by removing all chambers at non-maximal distance from a given element or flag. I consider a number of special cases of the above construction chosen among those which most frequently appear in the literature, proving that the resulting geometry is always simply conne...

We generalize the theory of sheaves to chamber systems. We prove that, given a chamber system C and a family R of proper residues of C containing all residues of rank c1, every sheaf defined over R admits a completion which extends C. We also prove that, under suitable hypotheses, a sheaf defined over a truncation of C can be extended to a sheaf fo...

Many constructions in diagram geometry exploit relations bearing some resemblance with the parallelism of an affine space. An abstract definition of parallelism with a discussion of two of those constructions (namely gluing and parallel expansion) has been given in a paper by F. Buekenhout, C. Huybrechts and A. Pasini (Parallelism in diagram geomet...

Some old and new constructions of the tilde geometry (the flag transitive con-nected triple cover of the unique generalized quadrangle W(2) of order (2, 2)) are discussed. Using them, we prove some properties of that geometry. In particular, we compute its gen-erating rank, we give an explicit description of its universal projective embedding and w...

An n-fullerene is an n-dimensional cell complex where the stars of the points are (n−1)-dimensional simplices and the 2-cells are pentagons or hexagons. We say that an n-fullerene is uniform if the number of hexagonal faces containing a given vertex p (and, when n>3, contained in a given 3-face X on p) does not depend on the choice of p (and X). Fo...

Given a set of types I, a type 0∈I, a subset J of I containing 0, and a diagram I0 over I\{0}, a geometry Γ over the set of types J is said to be locally truncated of I0-type if the J\{0}-residues of Γ are truncations of geometries or chamber systems belonging to I0. We give a sufficent condition for such a geometry to be the J-truncation of a cham...

In 4 we have studied the semibiplanes Σm,he=Af(Sm,he) obtained as affine expansions of the d -dimensional dual hyperovals of Yoshiara 6. We continue that investigation here, but from a graph theoretic point of view. Denoting byΓm, he the incidence graph of (the point-block system of)Σm, he, we prove that Γm,heis distance regular if and only if eith...

Each of the d -dimensional dual hyperovalsSmh discovered by Yoshiara 20 gives rise, via affine expansion, to a flag-transitive semibiplane Af(Smh). We prove that, if m + h = d + 1, thenAf (Smh) is an elation semibiplane. In the other cases, ifd > 2 then Af(Smh) is not isomorphic to any of the examples we are aware of, except possibly for certain se...

Let Δ be a finite thick dual polar space of rank 3. We say that a hyperplane H of Δ is locally singular (respectively, quadrangular or ovoidal) if H∩Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of Δ. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is loca...

Suppose Г is a Lie incidence geometry defined over some field F having a Lie incidence geometry Г0 of the same type but defined over a subfield F0 ≤ F as a subgeometry. We investigate the following question: how many points (if any at all) do we have to add to the point-set of Г0 in order to obtain a generating set for Г? We note that if Г is gener...

We consider flag-transitive L.L*-geometries where the residues of the planes are finite non-degenerate projective spaces of dimension at least 3 and the residues of the points are their duals. We prove that every geometry as above is a truncation of a building of type D
n
, for a suitable n depending on the orders of that geometry.

This very short note is a concise presentation of one of the many ways to approach diagram geometry. I wrote it in 1999 under request of a colleague, for Encyclopedia of Mathematics, an online encyclopedia first published in 2001.

We survey the known extended generalized quadrangles with point-residues of order (q−1,q+1) and (q+1,q−1) and construct a new infinite family of order (q+1,q−1) (q odd).

Let H be a geometric hyperplane of a classical finite generalized quadrangle Q and let C = Q \ H be its complement in Q, viewed as a point-line geometry. We shall prove that C admits a flag-transitive automorphism group if and only if H spans a hyperplane of the projective space in which Q is naturally embedded (but with Q viewed as Q(4, q) when Q...

We construct a simply connected flag-transitive circular extension of the dual affine space AG*(3,4), with 2^{13}:3.M_{22}2 as full automorphism Group and we show that this geometry as well as iits unique flag-transitive proper quotient are the unique flag-transitive circular extensions of finite thich dual affine spaces whose full automorphism Gro...

A C2.L-geometry is a geometry of rank 3 with elements called points, lines and quads, where residues of points are linear spaces, residues of lines are generalized digons and residues of quads are generalized quadrangles. Some sufficient conditions can be found in the literature for a C2.L-geometry to be a quotient of a truncated Cn-building. We sh...

We classify the flag-transitive circular extensions of line-point systems of finite projective geometries.

We characterizeC2.c-geometries that are truncations of almost-thinCn-geometries andC2.c-geometries covered by truncated almost-thin buildings of typeCn. Then we show how to profit from those characterizations in the investigation of a number of special cases. The proof of our main theorem is a rearrangement of the proof of a theorem by Brouwer on r...

This paper is developed toI
2(2g).c-geometries, namely, point-line-plane structures where planes are generalized 2g-gons with exactly two lines on every point and any two intersecting lines belong to a unique plane.I
2(2g).c-geometries appear in several contexts, sometimes in connection with sporadic simple groups. Many of them are homomorphic imag...

A C_2.L-geometry is a geometry of rank 3 with elements called points, lines and quads, where residues of points are linear spaces, residues of lines are generalized digons and residues of quads are generalized quadrangles. Some sufficient conditions can be found in the literature for a C_2.L-geometry to be a quotient of a truncated C_n-building. We...

A c.U*-geometry is a geometry over the diagram c.L*, the point residues of which are finite dual unitals. Only one flag-transitive example is known. Its full automorphism group is Aut(J
2), but J2 also acts flag-transitively on it. We shall prove that this geometry is indeed the unique flag-transitive c.U*-geometry, thus obtaining a new geometric c...

## Questions

Questions (3)

Suppose you have an account in a social or professional network (e.g. RG). Next, suppose you die. This is not a wish, of course; but you know, it can happen. How can the administrators of the network get acquainted with the sad news? I am afraid that they will keep your account open for ever; even uploading some contributions of yours from times to times (e.g., a preprint, in RG), let you be happy with this or not, as if you were not even free to pass away.

Here is the question: is there any way to avoid slightly grotesque accidents like these?

Why this question? Here is a motivation: from times to times RG announces that someone, who passed away months or years ago, has just uploaded a paper of his. I know that this happens because one of his former coauthors has eventually decided to upload that paper. Nevertheless, it always shocks me a bit.

I recall that, given two totally ordered sets (A,<) and (B,<), an embedding of the first one into the latter is an injective mapping f from A into B such that, for any x, y in A, if x < y then f(x) < f(y). I guess that this question has already been asked and answered long ago. I have looked for an answer in a few books of set theory, but I couldn't find any. Here is a motivation of this question:

consider the family of isomorphism classes of total orders that can be defined on a given set U. If the answer to my question is affirmative, then a partial ordering is naturally defined on this family and more questions can thus be asked. For instance: does it admit minimal elements? If it does, are these minimal elements necessarily (isomorphism classes of) well orderings?