## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 10, Number 2 (2003), 231-261.

### Quasi-subgeometry partitions of projective spaces

#### Abstract

The notion of a subgeometry partition of a finite projective space $% PG(2m-1,q^{2})$ by $PG(m-1,q^{2})$'s and $PG(2m-1,q)$'s or a partition of $% PG(2m,q^{2})$ by $PG(2m,q)$'s is generalized to quasi-subgeometry partitions of $PG(2m-1,q^{d})$ by $PG(dm/e-1,q^{e})$'s for a set of divisors $e$ of $d$ and, partitions of $PG(2m,q^{2d})$ by $PG(d(2m+1)/f-1,q^{f})$'s for a set of divisors $f$ of $d$. In all cases, there are associated vector space spreads that are unions of `fans'.

More generally, in the arbitrary dimensional case, a complete theory of quasi-subgeometry partitions of $PG(V-1,D)$ corresponding to generalized spreads admitting $D^{\ast }$ as a fixed-point-free collineation group is obtained. When $D$ is a quadratic extension of a base field, `subgeometry' partitions are obtain.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 10, Number 2 (2003), 231-261.

**Dates**

First available in Project Euclid: 5 June 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1054818026

**Digital Object Identifier**

doi:10.36045/bbms/1054818026

**Mathematical Reviews number (MathSciNet)**

MR2015201

**Zentralblatt MATH identifier**

1045.51005

**Subjects**

Primary: 51E23: Spreads and packing problems

Secondary: 51A40: Translation planes and spreads

**Keywords**

quasi-subgeometry partition fans André Bruck-Bose

#### Citation

Johnson, Norman. Quasi-subgeometry partitions of projective spaces. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), no. 2, 231--261. doi:10.36045/bbms/1054818026. https://projecteuclid.org/euclid.bbms/1054818026