Desarguesian spreads and field reduction for elements of the semilinear group

The goal of this note is to create a sound framework for the interplay between field reduction for finite projective spaces, the general semilinear groups acting on the defining vector spaces and the projective semilinear groups. This approach makes it possible to reprove a result of Dye on the stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend it to P{\Gamma}L(n, q). Moreover a result of Drudge [5] relating Singer cycles with Desarguesian spreads, as well as a result on subspreads (by Sheekey, Rottey and Van de Voorde [19]) are reproven in a similar elementary way. Finally, we try to use this approach to shed a light on Condition (A) of Csajbok and Zanella, introduced in the study of linear sets [4]

Hennebique’s journal 'Le Béton armé': a close reading of the genesis of concrete construction in Belgium

In June 1898, François Hennebique issued the monthly journal Le Béton Armé. Published until 1939, with 378 issues in all, this platform on the interface between information and propaganda serves as a perfect means to obtain a comprehensive overview of Hennebique’s legacy. Giving an insight into the increasing sphere of action, the growing number of applications and the hierarchic structure and policy of the firm, the journal is a work of reference, essential to document the unremitting development of concrete construction. By means of a close reading (based on the collections preserved at Ghent University and the Centre d’archives du XXe siècle de l’ifa in Paris), the content, meaning, and changing discourse of Le Béton Armé will be critically analyzed. Fitting within the scope of a PhD on the history of concrete construction in Belgium (www.architecture.ugent.be/concrete), particular attention will be given to the application of ‘le système Hennebique’ in Belgium

On sets without tangents and exterior sets of a conic

A set without tangents in \PG(2,q) is a set of points S such that no line meets S in exactly one point. An exterior set of a conic $\mathcal{C}$ is a set of points \E such that all secant lines of \E are external lines of $\mathcal{C}$. In this paper, we first recall some known examples of sets without tangents and describe them in terms of determined directions of an affine pointset. We show that the smallest sets without tangents in \PG(2,5) are (up to projective equivalence) of two different types. We generalise the non-trivial type by giving an explicit construction of a set without tangents in \PG(2,q), $q=p^h$, $p>2$ prime, of size $q(q-1)/2-r(q+1)/2$, for all $0\leq r\leq (q-5)/2$. After that, a different description of the same set in \PG(2,5), using exterior sets of a conic, is given and we investigate in which ways a set of exterior points on an external line $L$ of a conic in \PG(2,q) can be extended with an extra point $Q$ to a larger exterior set of $\mathcal{C}$. It turns out that if $q=3$ mod 4, $Q$ has to lie on $L$, whereas if $q=1$ mod 4, there is a unique point $Q$ not on $L$

A small minimal blocking set in PG(n,p^t), spanning a (t-1)-space, is linear

In this paper, we show that a small minimal blocking set with exponent e in PG(n,p^t), p prime, spanning a (t/e-1)-dimensional space, is an F_p^e-linear set, provided that p>5(t/e)-11. As a corollary, we get that all small minimal blocking sets in PG(n,p^t), p prime, p>5t-11, spanning a (t-1)-dimensional space, are F_p-linear, hence confirming the linearity conjecture for blocking sets in this particular case

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/4

Pseudo-ovals in even characteristic and ovoidal Laguerre planes

Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set $\mathcal{A}$ of $(n-1)$-spaces in $\mathrm{PG}(3n-1,q)$ such that any three span the whole space. Pseudo-arcs of size $q^n+1$ are called pseudo-ovals, while pseudo-arcs of size $q^n+2$ are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in $\mathrm{PG}(2,q^n)$. We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in $\mathrm{PG}(3n-1,q)$, where $q$ is even and $n$ is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes

Characterisations of elementary pseudo-caps and good eggs

In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $\mathrm{PG}(4n-1, q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(\mathcal{O})$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw