## Innovations in Incidence Geometry

### New results on covers and partial spreads of polar spaces

#### Abstract

We investigate blocking sets of projective spaces that are contained in cones over quadrics of rank two. As an application we obtain new results on partial ovoids, partial spreads, and blocking sets of polar spaces. One of the results is that a partial ovoid of $H(3,q2)$ with more than $q3−q+1$ points is contained in an ovoid. We also give a new proof of the result that a partial spread of $Q(4,q)$ with more than $q2−q+1$ lines is contained in a spread; this is the first common proof for even and odd $q$. Finally, we improve the lower bound on the size of a smallest blocking set of the symplectic polar space $W(3,q)$, $q$ odd.

#### Article information

Source
Innov. Incidence Geom., Volume 1, Number 1 (2005), 19-34.

Dates
Accepted: 20 January 2005
First available in Project Euclid: 26 February 2019

https://projecteuclid.org/euclid.iig/1551206814

Digital Object Identifier
doi:10.2140/iig.2005.1.19

Mathematical Reviews number (MathSciNet)
MR2213952

Zentralblatt MATH identifier
1116.51010

#### Citation

Klein, Andreas; Metsch, Klaus. New results on covers and partial spreads of polar spaces. Innov. Incidence Geom. 1 (2005), no. 1, 19--34. doi:10.2140/iig.2005.1.19. https://projecteuclid.org/euclid.iig/1551206814

#### References

• A. Aguglia, G. L. Ebert and D. Luyckx, On partial ovoids of hermitian surfaces, Bulletin Belg. Math. Soc. | Simon Stevin, To appear.
• A. Bichara and G. Korchmáros, Note on $(q+2)$-sets in a Galois plane of order $q$, Ann. Discrete Math. 14 (1982), 117–122.
• M. R. Brown, J. De Beule and L. Storme, Maximal partial spreads of $T_2({\cal O})$ and $T_3({\cal O})$, European J. Combin. 24 (2004), 73–84.
• G. Ebert, Personal communication.
• J. Eisfeld, L. Storme, T. Szőnyi and P. Sziklai, Covers and blocking sets of classical generalized quadrangles, Discrete Math. 238 (2001), 35–51.
• P. Govaerts, Ph.D. Thesis, Ghent University, Belgium, Gent, 2003.
• G. Tallini, Blocking sets with respect to planes in ${\rm PG}(3,q)$ and maximal spreads of a nonsingular quadric in ${\rm PG}(4,q)$, In: Proceedings of the First International Conference on Blocking Sets (Giessen, 1989) (1991), 141–147.