Bilinear flocks

α-Flokki and Partial α-Flokki

“…In this section, we connect α-flokki translation planes and maximal α-partial spreads (which are injective but not bijective) with flocks of the cone C α . The ideas presented here originate from Cherowitzo-Holder [4], and Kantor-Penttila [12], in the finite case. Definition 3.1.…”

“…Problem (2) is answered in the negative by the in-and out-star bilinear flokki of Cherowitzo and Holder [4] (e.g. Examples 1 and 2).…”

“…Problem (4). In Kantor and Penttila, it is noted that if g, f give a flokki plane then gf −1 , f −1 also give a flokki plane.…”

“…A result of Thas [15] shows that flocks of quadratic cones whose planes share a point must be linear in the even characteristic case and either linear or Knuth-Kantor in odd characteristic and hence bilinear flocks of quadratic cones with intersecting supporting lines do not exist in the finite case. Also, no bilinear flocks of quadratic cones with skew supporting lines are known in the finite case (see [4] for a more thorough study of finite bilinear flocks). However, Biliotti and Johnson [1], show that bilinear flocks can exist in PG(3, K), where K is a infinite field.…”

“…In fact, the situation is much more complex for infinite flocks, for example, there are n-linear flocks for any positive integer n (the planes share exactly n lines). Recently, Cherowitzo and Holder [4], found an extremely interesting bilinear q-flokki using ideas from blocking sets. It might be suspected that the Cherowitzo-Holder q-flokki might be algebraically lifted from a translation plane, and indeed, in this article, we show that this is, in fact, the case.…”