Sara Rottey

The isomorphism problem for linear representations and their graphs The isomorphism problem for linear representations and their graphs

Abstract A linear representation T*n(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1; q), where K is contained in a hyperplane. We put constraints on K which ensure that every automorphism of T*n(K) is induced by a collineation of the ambient projective space. This allows us to show that, under certain conditions, two linear representations T*n(K) and T*n(K′) are isomorphic if and only if the point sets K and K′ are PΓL-equivalent. We also deal with the slightly more general problem of isomorphic incidence graphs of linear representations. In the last part of this paper, we give an explicit description of the group of automorphisms of T*n(K) that are induced by collineations of PG(n + 1; q).

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 A of ( n - 1 ) -spaces in PG ( 3 n - 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 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 PG ( 3 n - 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.

Blocking sets of the classical unital

It is known that the classical unital arising from the Hermitian curve in PG(2,9) does not have a 2-coloring without monochromatic lines. Here we show that for q≥4 the Hermitian curve in PG(2,q2) does possess 2-colorings without monochromatic lines. We present general constructions and also prove a lower bound on the size of blocking sets in the classical unital.

Maximal partial line spreads of non-singular quadrics

For

Unitals with many Baer secants through a fixed point

n \ge 9

Subgeometries in the André/Bruck-Bose representation

, we construct maximal partial line spreads for non-singular quadrics of

Characterisations of elementary pseudo-caps and good eggs

PG(n,q)