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 $L$

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]

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

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

Exploration of the influence of 5-iodo-2'-deoxyuridine incorporation on the structure of d[CACG(IDU)G]

The first antiviral nucleoside 5-iodo-2'-deoxyuridine (IDU) against herpes simplex virus type 1 and type 2 is a thymidine analogue, i.e. the C5 methyl group is replaced by an I atom. The structure of the self-complementary hexamer d[CACG(IDU)G] was determined by single-crystal X-ray diffraction techniques. The orthorhombic crystals belong to space group P2(1)2(1)2(1), with unit-cell parameters a = 18.16, b = 30.03, c = 41.99 Angstrom. Refinement in the resolution range 20 - 1.3 Angstrom converged with a final R1 = 0.167, including 43 water molecules and two cobalt hexammine complexes. The incorporation of a large I atom has only minor consequences for the overall structure as is noticed in the IDU . A base pairs, which are of the common Watson - Crick type. To contribute to the still puzzling mechanism of this historically important agent, details of base stacking, helical parameters, hydration etc. have been studied. A general scheme of cobalt hexammine-binding modes in Z-DNA is provided, revealing similar binding modes for the reported structure

Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields