## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 11, Number 2 (2004), 281-288.

### Quadratic sets of a $3$-dimensional locally projective regular planar space

Roberta Di Gennaro and Nicola Durante

#### Abstract

In this paper quadratic sets of a $3$-dimensional locally projective regular planar space $(\cal S,\cal L,\cal P)$ of order $n$ are studied and classified. It is proved that if in $(\cal S,\cal L,\cal P)$ there is a non-degenerate quadratic set $\bf H$, then the planar space is either $\mathop{\rm{PG}}(3,n)$ or $\mathop{\rm{AG}}(3,n)$. Moreover in the first case $\bf H$ is either an ovoid or an hyperbolic quadric, in the latter case $\bf H$ is either a cylinder with base an oval or a pair of parallel planes.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 11, Number 2 (2004), 281-288.

**Dates**

First available in Project Euclid: 11 June 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1086969318

**Digital Object Identifier**

doi:10.36045/bbms/1086969318

**Mathematical Reviews number (MathSciNet)**

MR2080428

**Zentralblatt MATH identifier**

1081.51006

#### Citation

Di Gennaro, Roberta; Durante, Nicola. Quadratic sets of a $3$-dimensional locally projective regular planar space. Bull. Belg. Math. Soc. Simon Stevin 11 (2004), no. 2, 281--288. doi:10.36045/bbms/1086969318. https://projecteuclid.org/euclid.bbms/1086969318