Lower bounds for the size of a complete partial ovoid in a non-degenerate Hermitian surface are obtained. For even characteristic, a sharp bound is obtained and all examples of this size are described. Next, a general construction method for locally hermitian partial ovoids is explained, which leads to interesting small examples. Finally, a conjecture is given for the size of the largest complete strictly partial ovoid. By using partial derivation, several examples of complete strictly partial ovoids of this size are provided.

In this paper we determine the minimal blocking sets of chords of an irreducible conic $\mathcal C$ in the desarguesian projective plane $PG(2,q)$, $q$ even. Similar results on blocking sets of external lines, as well as of nonsecant lines, are given in [1], [3], and [2].

Using Grundhöfer's construction of the Figueroa planes from Pappian planes which have an order three planar collineation ${\widehat \alpha }$, we show that any Figueroa plane (finite or infinite) of characteristic not two cannot have a Fano subplane which includes an ${\widehat \alpha }$ invariant triangle.

An arc in a projective plane whose secants meet some exterior line in the minimum number of points is said to be hyperfocused on that line. This is similar to a concept introduced by Simmons in the design of a geometric secret sharing scheme. Drake and Keating have recently rediscovered the idea under the guise of hyperovals in nets. We will answer two open questions raised by Drake and Keating. We also provide a classification of small hyperfocused arcs in Desarguesian planes using the graph-theoretic concept of 1-factorizations of complete graphs.

We show that the strongly regular graph with parameters $v=105$, $k=32$, $\lambda=4$ and $\mu=12$ is uniquely determined by its parameters (upto isomorphism).

In this paper, the non-existence of ovoids of the polar space $H(5,4)$ is shown using a geometrical and combinatorial approach. We also give a new and unified proof for the non-existence of ovoids in the polar spaces $Q^-(2n+1,q)$, $H(2n,q^2)$ and $W(2n+1,q)$ for $n\geq 2$.

We describe the two smallest minimal blocking sets of ${\rm Q}(2n,3)$, $n\geqslant 3$. To obtain these results, we use the characterization of the smallest minimal blocking sets of ${\rm Q}(6,3)$, different from an ovoid. We also present some geometrical properties of ovoids of ${\rm Q}(6,q)$, $q$ odd.

We construct new examples of sets of points on the Klein quadric $\mathcal{K}$, $q$ even, having exactly two intersection sizes 0 and $\alpha$ with lines on $\mathcal{K}$. By the well-known Plücker correspondence, these examples yield new $(0,\alpha)$-geometries embedded in $PG(3,q)$, $q$ even.

We classify all inclusions of irreducible spherical buildings of equal rank~$\geq~3$, with the only restriction that the characteristic of the defining field is different from $2$ if there are residues which are Moufang quadrangles.

Semipartial geometries (SPG) were introduced in 1978 by Debroey and Thas. As some of the examples they provided were embedded in affine space it was a natural question to ask whether it was possible to classify all SPG embedded in affine space. In $AG(2,q)$ and $AG(3,q)$ a complete classification was obtained. Later on it was shown that if an SPG, with $\alpha>1$, is embedded in affine space it is either a linear representation or $\mathrm{TQ}(4,2^h)$. In this paper we derive general restrictions on the parameters of an SPG to have a linear representation and classify the linear representations of SPG in $AG(4,q)$, hence yielding the complete classification of SPG in $AG(4,q)$, with $\alpha>1$.

In this paper, we give an alternative construction of the Hölz design $D_{Hölz}(q)$, for $q\not\equiv 2$~mod~$3$. If $q\equiv 2$~mod~$3$, then our construction yields a $2-(q^3+1,q+1,\frac{q+4}{3})$-subdesign of the Hölz-design. The construction uses two hexagons embedded in the parabolic quadric $Q(6,q)$.

Cameron-Liebler line classes are sets of lines in PG(3,q) that contain a fixed number $x$ of lines of every spread. Cameron and Liebler classified them for $x\in\{0,1,2,q^2-1,q^2,q^2+1\}$ and conjectured that no others exist. This conjecture was disproved by Drudge and his counterexample was generalised to a counterexample for any odd $q$ by Bruen and Drudge. In this paper, we give the first counterexample for even $q$, a Cameron-Liebler line class with parameter $7$ in PG(3,4). We also prove the nonexistence of Cameron-Liebler line classes with parameters $4$ and $5$ in PG(3,4) and give some properties of a hypothetical Cameron-Liebler line class with parameter $6$ in PG(3,4).

Recently, we showed that any $3$-Moufang generalized quadrangle is automatically a Moufang quadrangle. In another recent paper, Katrin Tent \cite{Ten:04a} borrowed an argument of the second author to show that the half Moufang condition implies the Moufang condition for generalized quadrangles. In the present paper we show that this argument can be used to further weaken the hypotheses: we define the half $3$-Moufang condition as a kind of greatest common divisor of the $3$-Moufang condition and the half Moufang condition and show that it implies the Moufang condition.

The set of translation planes with spreads in $PG(3,q)$ admitting cyclic affine homology groups of order $q+1$ is shown to be equivalent to the set of flocks of quadratic cones in $PG(3,q)$. The analysis is general and considers analogous homology groups in $PG(3,K)$, for $K$ an arbitrary field and corresponding partial flocks of quadratic cones in $PG(3,K)$.

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 morphism of embeddings $\tilde{f}:\tilde{e}\rightarrow e$ and, for every embedding $e':\Gamma\rightarrow PG(V')$ with $V'$ defined over $K$, if there is a morphism $g:e'\rightarrow e$ then a morphism $f:\tilde{e}\rightarrow e'$ also exists such that $\tilde{f} = gf$. If $e = \tilde{e}$ then we say that $e$ is {\em dominant}. Clearly, hulls are dominant. Let now $\Gamma$ be a non-degenerate polar space of rank $n\geq 3$. We shall prove the following: A lax embedding $e:\Gamma\rightarrow PG(V)$ is dominant if and only if, for every geometric hyperplane $H$ of $\Gamma$, $e(H)$ spans a hyperplane of $PG(V)$. We shall also give some applications of the above result.

It is shown that every automorphism of a classical unital over a certain (not necessarily commutative) field is induced by a semi-similitude of a corresponding hermitian form. In particular, this is true if the form uses an involution of the second kind.

In this paper, I will start with posing three fundamental and old questions on (elation) generalized quadrangles, and survey tersely answers on these questions coming from recent work of S. E. Payne and the author of this paper. I will then introduce a fourth question posed recently by S. E. Payne, and will provide a general answer to this question, a result independently obtained by R. Rostermundt for the Hermitian quadrangles $H(3,q^2)$, $q$ even, in an entirely different fashion. Finally, I will show that this answer yields examples of elation generalized quadrangles for which the automorphism group fixing the elation point is not induced by the automorphisms of the elation group fixing the associated $4$-gonal family.

We settle two problems posed by the first author concerning the Biggs-Smith geometry. The first is to determine the generating rank, the second is to prove that this geometry possesses exactly eighteen ovoids.

A notion of dimensional dual polar spaces of polar type is given with a family of new examples. Nonexistence is shown for those of dimension $3$, while there are just two isomorphism classes for those of dimension $2$.