Inherited conics in Hall planes

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic

The classification of inherited hyperconics in Hall planes of even order

AbstractIn this note we complete the classification of inherited hyperconics in Hall planes of even order that was started by O’Keefe and Pascasio by proving that in the cases left open in [C.M. O’Keefe, A.A. Pascasio, Images of conics under derivation, Discrete Math. 151 (1996) 189–199] there are no inherited hyperconics

Some sporadic translation planes of order 11211^2

In \cite{PK}, the authors constructed a translation plane $\Pi$ of order $11^2$ arising from replacement of a sporadic chain $F'$ of reguli in a regular spread $F$ of $PG(3,11)$. They also showed that two more non isomorphic translation planes, called  $\Pi_1$ and $\Pi_{13}$, arise respectively by derivation and double derivation in $F\setminus F'$ which correspond to a further replacement of a regulus with its opposite regulus and a pair of reguli with their opposite reguli, respectively.  In \cite{AL}, the authors proved that the translation complement of $\Pi$ contains a subgroup isomorphic to \SL(2,5). Here, the full collineation group of each of the planes $\Pi$, $\Pi_1$ and $\Pi_{13}$ is determined

Elation generalised quadrangles of order (s,p), where p is prime

We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle).Comment: 14 page