Luyckx, Deirdre

Thas, Joseph

English

Ghent University Academic Bibliography

1-systems of Q(6, q) and Q−(7, q): recent resultsDeirdre Luyckx 1 Joseph A. ThasGhent UniversityDepartment of Pure Mathematics and Computer AlgebraGalglaan 2, B-9000 GhentBelgiumdluyckx@cage.rug.ac.bejat@cage.rug.ac.beIn [2], m-systems of polar spaces were introduced by Shult and Thas anda description of several classes of examples is given. However, few other ex-amples are known until now. First, we briefly discuss the construction of anew class of 1-systems of Q(6, q), q odd, starting from a generalization of aflock of a quadratic cone in PG(3, q). Secondly, we mention a uniquenessresult about 1-systems of the quadric Q−(7, q).An i-flock of a quadratic cone LQ(2, q) with line vertex in PG(4, q), qodd, is defined as a partition of LQ(2, q)\L in q2 mutually disjoint conics suchthat the planes of the elements of the i-flock pairwise intersect in internalpoints of LQ(2, q). It can be shown that to every i-flock of LQ(2, q) a locallyhermitian 1-system of Q(6, q) is associated and conversely.Applying the theory of the i-flocks to the semi-classical non-hermitianspread S[9] of the hexagon H(q), q odd and q ≡ 1 mod 3 (see [1]), which islocally hermitian at some line L, we find an interesting geometric constructionof S[9] starting from a rational normal cubic scroll R3 having L as directrixline. Apparently the conics on R3 determine the q2 conic planes of the i-flock associated with S[9] and hence the i-flock can be reconstructed fromthe cubic scroll. Surprisingly this geometric construction not only yields the1-system S[9]; it turns out that different cubic scrolls may give rise to non-isomorphic locally hermitian 1-systems of Q(6, q). In particular, there are(q − 3)/2 orbits in the set of all non-hermitian locally hermitian 1-systemsof Q(6, q) constructed from a cubic scroll, under the subgroup of PGL(7, q)fixing Q(6, q).1The first author is Research Assistant of the Fund for Scientific Research – Flanders(Belgium) (F.W.O.)1It can be shown that a locally hermitian non-hermitian 1-system ofQ(6, q),q odd, is semi-classical if and only if it arises from a rational normal cubicscroll R3 with directrix line L ⊆ Q(6, q) and with the property that all pointsof R3 \L are internal points of Q(6, q). As it is possible to determine all suchcubic scrolls, this yields a complete characterization and determination ofthe locally hermitian semi-classical 1-systems of Q(6, q) for q odd.Now, let M be a 1-system of Q−(7, q). Then it can be shown that everyline of Q−(7, q) contains 0, 1, 2 or q + 1 points on lines of M and that thelatter holds if and only if the line itself belongs toM. For q odd, this impliesthat every plane of Q−(7, q) either contains a line of M or an irreducibleconic of points on lines of M. This observation enabled us to prove thatQ−(7, q) has a unique 1-system for q odd, up to a projectivity. This is animportant result because 1-systems of Q−(7, q) were candidates to yield newgeneralized quadrangles, as described in [3].References[1] I. Bloemen, J. A. Thas and H. Van Maldeghem. Translation ovoids of generalizedquadrangles and hexagons.Geom. Dedicata, 72(1):19-62, 1998.[2] E. E. Shult and J. A. Thas. m-systems of polar spaces.J. Combin. Theory Ser. A, 68(1):184-204, 1994.[3] E. E. Shult and J. A. Thas. Constructions of polygons from buildings.Proc. London Math. Soc. (3), 71(2):397-440, 19952

1-systems of Q(6,q) and Q-(7,q): recent results

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1-systems of Q(6,q) and Q-(7,q): recent results

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