## Innovations in Incidence Geometry

- Innov. Incidence Geom.
- Volume 6-7, Number 1 (2007), 295-305.

### Locally hermitian partial ovoids of unitary polar spaces and partial ovoids of orthogonal polar spaces

#### Abstract

In order to study unitals in the projective plane $PG\left(2,{q}^{2}\right)$, F. Buekenhout gave a representation in $PG\left(4,q\right)$ of the unitary polar space $H\left(2,{q}^{2}\right)$ as points of a quadratic cone on a ${Q}^{-}\left(3,q\right)$.

G. Lunardon used the Barlotti-Cofman representation of $PG\left(3,{q}^{2}\right)$ to represent $H\left(3,{q}^{2}\right)$ in $PG\left(6,q\right)$ as a cone on a ${Q}^{+}\left(5,q\right)$. He also proved that to any locally hermitian ovoid of $H\left(3,{q}^{2}\right)$ corresponds an ovoid of ${Q}^{+}\left(5,q\right)$ and conversely.

In this paper, we study the Barlotti-Cofman representation of the unitary polar space $H\left(n,{q}^{2}\right)$ for all $n$ and we prove that to any locally hermitian partial ovoid of such spaces corresponds a partial ovoid of an orthogonal polar space, and conversely. Further the locally hermitian partial ovoid is maximal if and only if the corresponding partial ovoid of the orthogonal polar space is maximal. As a consequence of the previous connection and a result of A. Klein we obtain a geometric proof to derive that the orthogonal polar space ${Q}^{+}\left(4n+1,q\right)$ has no ovoid when $n>{q}^{3}$.

#### Article information

**Source**

Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 295-305.

**Dates**

Received: 10 January 2008

First available in Project Euclid: 28 February 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.iig/1551323185

**Digital Object Identifier**

doi:10.2140/iig.2008.6.295

**Mathematical Reviews number (MathSciNet)**

MR2515273

**Zentralblatt MATH identifier**

1175.51005

**Subjects**

Primary: 05B25: Finite geometries [See also 51D20, 51Exx] 51E20: Combinatorial structures in finite projective spaces [See also 05Bxx]

**Keywords**

polar spaces partial ovoids ovoids

#### Citation

Siciliano, Alessandro. Locally hermitian partial ovoids of unitary polar spaces and partial ovoids of orthogonal polar spaces. Innov. Incidence Geom. 6-7 (2007), no. 1, 295--305. doi:10.2140/iig.2008.6.295. https://projecteuclid.org/euclid.iig/1551323185