Results on partial geometries with an abelian Singer group of rigid type

TL;DR

This paper investigates partial geometries with an abelian Singer group of rigid type, showing their rarity below one million points and excluding many promising parameter sets, culminating in a conjecture about their existence.

Contribution

It proves that partial geometries of rigid type with fewer than one million points are limited to known or hypothetical cases and rules out many potential parameter sets.

Findings

Partial geometries of rigid type are very rare below 1,000,000 points.

Excluded an infinite set of promising parameters for such geometries.

Identified specific known and hypothetical geometries fitting the criteria.

Abstract

A partial geometry $S$ admitting an abelian Singer group $G$ is called of rigid type if all lines of $S$ have a trivial stabilizer in $G$. In this paper, we show that if a partial geometry of rigid type has fewer than $1000000$ points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.