Flocks and partial flocks of hyperbolic quadrics via root systems
A flock of the hyperbolic quadric \(Q^+(3,q)\) in \(PG(3,q)\) is a partitioning of the point set of this quadric into \(q+1\) pairwise disjoint conics. The flocks of the hyperbolic quadric have been classified by \textit{L. Bader } and \textit{G. Lunardon} [Geom. Dedicata 29, 177-183 (1989; Zbl 0673.51010)] and the classification result shows that there are three sporadic examples occurring for \(q=11, 23\) and \(59\). A fundamental result of \textit{J. A. Thas} [Contemp. Math. 111, 187-218 (1990; Zbl 0728.51010)] shows that for any flock of the hyperbolic quadric and for any plane of that flock, the reflection about that plane stabilising the hyperbolic quadric also stabilises the flock. Since also the root systems arise from reflection groups, there is the possibility of a link between the exceptional flocks and root systems in \(R^4\). The authors show that there effectively is a link by constructing infinite families of partial flocks of sizes 12, 24 and 60 of the hyperbolic quadric of \(PG(3,q)\), for \(q\) congruent to \(-1\) modulo 12, 24, 60 respectively, from the root systems of type \(D_4, F_4, H_4\) respectively. The smallest member of each of these families is an exceptional flock. The authors then characterise these partial flocks in terms of the rectangle condition of Benz, and by not being subflocks of linear flocks or of Thas flocks. And by using a matrix model for \(PG(3,q)\), they give an explicit correspondence between the regular subgroups of \(PGL(2,q)\) and the flocks of \(Q^+(3,q)\), showing that each of the partial flocks of \(Q^+(3,q)\) they constructed corresponds to a semiregular subgroup of \(PGL(2,q)\) isomorphic to \(A_4, S_4\) or \(A_5\).
scientific article; zbMATH DE number 1879607
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