#### Vol. 6, No. 1, 2008

 Recent Issues Volume 17, Issue 3 (189-249) Volume 17, Issue 2 (77-188) Volume 17, Issue 1 (1-75) Volume 16, Issue 1 Volume 15, Issue 1 Volume 14, Issue 1 Volume 13, Issue 1 Volume 12, Issue 1 Volume 11, Issue 1 Volume 10, Issue 1 Volume 9, Issue 1 Volume 8, Issue 1 Volume 6+7, Issue 1 Volume 5, Issue 1 Volume 4, Issue 1 Volume 3, Issue 1 Volume 2, Issue 1 Volume 1, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN (electronic): 2640-7345 ISSN (print): 2640-7337 Author Index To Appear Other MSP Journals
Locally hermitian partial ovoids of unitary polar spaces and partial ovoids of orthogonal polar spaces

### Alessandro Siciliano

Vol. 6 (2008), No. 1, 295–305
DOI: 10.2140/iig.2008.6.295
##### 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 $n>{q}^{3}$.

##### Keywords
polar spaces, partial ovoids, ovoids
##### Mathematical Subject Classification 2000
Primary: 05B25, 51E20