Characterisations of elementary pseudo-caps and good eggs

TL;DR

This paper investigates the properties of pseudo-caps and eggs in projective geometry, extending previous results to larger classes and providing new characterizations in both odd and even characteristic fields.

Contribution

It extends the characterization of elementary eggs and good eggs, generalizing previous results to larger pseudo-caps and providing new geometric proofs.

Findings

A similar property for large pseudo-caps in odd and even characteristic fields.

Improved conditions for identifying elementary eggs in even characteristic.

A new geometric proof of Lavrauw's theorem.

Abstract

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.