Some results on Spreads and Ovoids

Rendiconti di Matematica, Serie VII Volume 30, Roma (2010), 23-32 Some results on Spreads and Ovoids LAURA BADER Dedicated to Professor Marialuisa de Resmini Abstract: We survey some results on ovoids and spreads of finite polar spaces, focusing on the ovoids of H(3, q 2 ) arising from spreads of P G(3, q) via indicator sets and Shult embedding, and on some related constructions. We conclude with a remark on symplectic spreads of P G(2n − 1, q). 1 – Introduction Let q be any prime power and let P G(2n − 1, q) be the projective space of dimension 2n − 1 over the Galois field GF (q). A (n − 1)-spread S of P G(2n − 1, q) is a set of q n + 1 mutually skew (n − 1)-dimensional subspaces; hence the elements of S partition the pointset of P G(2n − 1, q). Spreads of P G(2n − 1, q) define translation planes of order q n , with kernel containing GF (q), embedding P G(2n − 1, q) as a hyperplane in a P G(2n, q) and using the well known AndréBruck/Bose construction, and conversely. This relationship is probably the main motivation for the study of spreads, and the most studied case is n = 2. Bruck in [8] introduced indicator sets in finite desarguesian projective planes of square order, and their links with line spreads of projective 3-spaces have been studied in the next years by Bruck himself in [9] and by Bruen in [10]; a few years later, Lunardon in [15] further studied that relationship, mainly from Key Words and Phrases: Spread – Indicator set – Linear representation – Hermitian variety A.M.S. Classification: Primary 51A50, Secondary 51E20, 51A40. 24 LAURA BADER [2] the synthetic geometric point of view: with any spread of P G(3, q) a family of indicator sets is associated. Indicator sets have been somehow aside for many years, until Shult in [19] proved that a suitable set of lines, presently called a Shult set, defines a locally Hermitian ovoid of the Hermitian variety via the so-called Shult embedding, and conversely. As a Shult set is the point-line dual of an indicator set, there immediately followed a link between spreads of P G(3, q) and families of locally Hermitian ovoids of H(3, q 2 ), which was first studied by Cossidente, Ebert, Marino and Siciliano in [11] focusing on those associated with the regular spread, the so called classical and semiclassial ovoids of the Hermitian variety. In the subsequent paper [12] Cossidente, Lunardon, Marino and Polverino classified the ovoids arising from the regular spread and from a (proper) semifield spread via the above construction, while in [2] Bader, Marino, Polverino and Trombetti further studied the collineation group of the translation ovoids constructed via a Shult embedding and pointed out that two constructions which could be performed (a family of ovoids of the Klein quadric from the given family of locally Hermitian ovoids of the Hermitian variety via a construction of Lunardon [17] and a family of line spreads from the given family of Shult sets via a construction of Thas [21]) do not produce any new example. Here we deal with these results and we conclude the paper with a remark linking symplectic spreads of P G(2n − 1, q) and Thas maximal arcs in projective planes of order q n and kernel containing GF (q). 2 – Spreads of P G(3, q), ovoids of H(3, q 2 ) and some related constructions 2.1 – Spreads, indicator sets, Shult sets View Σ = P G(3, q) as a canonical subgeometry of a Σ∗ = P G(3, q 2 ); let σ be the collineation of Σ∗ fixing Σ pointwise (hence σ 2 = id) and let S be any spread of Σ. Fix a line l in S. A plane π ∼ = P G(2, q 2 ) of Σ∗ is an indicator plane of S if π ∩ Σ = l; the indicator set of S in π is Iπ (S) = {m∗ ∩ π|m ∈ S}, where m∗ denotes the unique line of Σ∗ containing m. The set Iπ (S) has size q 2 and none of its secants contains points of l; conversely, any set I  of points of π satisfying the previous two properties canonically defines a spread, namely S  = {< Q, Qσ > ∩Σ | Q ∈ I  } ∪ {l} and Iπ (S  ) = I  . Hence, with any spread S a family is associated of indicator sets Iπ (S). Furthermore, the spread S is regular if and only if any Iπ (S) is either an affine line (classical indicator set) or an affine Baer subplane (semiclassical indicator set). For more details, see e.g. [8], [9], [10] and [15]. Let Σ = P G(3, q), Σ∗ = P G(3, q 2 ), the plane π and the line l be as above, and denote by l∗ the line of Σ∗ containing l. Let π̂ be the dual plane of π and let P denote the point of π̂ corresponding to the line l∗ . The points of l are [3] Some results on Spreads and Ovoids 25 mapped to the lines of a cone ˆl of π̂ having vertex P . Let F be the set of lines of π̂ corresponding to the points of the indicator set I. Then: (i) π̂ is a projective plane with a distinguished degenerate Hermitian variety (the Baer subpencil ˆl with vertex P ); (ii) F is a set of q 2 lines of π̂, none of which contains P ; (iii) any two distinct lines of F intersect in a point not on the Baer subpencil. Any set of lines satisfying the above three properties is called a Shult set. Conversely, a Shult set defines, by any polarity of its plane, an indicator set. In conclusion, with any line spread a family of indicator sets or, equivalently, a family of Shult sets is associated. 2.2 – Shult embedding A Hermitian surface H = H(3, q 2 ) of P G(3, q 2 ) is the set of all isotropic points of a non-degenerate unitary polarity. A line of P G(3, q 2 ) meets H in 1 (tangent) or q + 1 (hyperbolic line) or q 2 + 1 (generator) points. The hyperbolic lines intersect H in Baer sublines which are called chords. An ovoid O of H is a set of q 3 + 1 points such that any generator of H contains exactly one point of O. The Hermitian curve H(2, q 2 ), intersection of H with any of its secant planes, is the classical ovoid. An ovoid is called locally Hermitian with respect to a point P if it is the union of q 2 chords of H through P and is called translation with respect to a point P if there is a collineation group of H fixing P , all the generators through P, and acting regularly on the points of O \ {P }. Note that any translation ovoid is locally Hermitian ([7]) but not conversely, and a classical ovoid of H is a translation ovoid with respect to each of its points. Start off with a spread S of P G(3, q), fix a line l in S, an indicator plane π through l as above, construct the indicator set and polarize to a Shult set F with respect to the subpencil ˆl in the plane π̂ = P G(2, q 2 ); embed the plane π̂ in a P G(3, q 2 ) containing a Hermitian surface H such that π̂ is the tangent plane to H at P and ˆl = H ∩ π̂ ; denote by  ρ be the polarity defined by H. Then Shult has proved in [19] that Oπ (S) = {Lρ |L ∈ F} is an ovoid of H, which is, by construction, locally Hermitian with respect to its point P . The above construction is presently called a Shult embedding following [11]. We explicitly note that on the other hand, via the so-called Hermitian embedding defined by Cossidente, Ebert, Marino and Siciliano in [11], symplectic spreads of (...truncated)