Homology groups of translation planes and flocks of quadratic cones, II; j-planes

Abstract

The set of j-planes with spreads in PG(3,K), for K a field admitting a quadratic field extension K⁺ is shown to be equivalent to the set of all det K⁺-monomial partial flocks of a quadratic cone. Using this connection, when K is GF(2^r), the set of j-planes is determined and j = 0, 1, or 2 and correspond to the linear, Walker/Betten or Payne conical flocks, respectively. When K is the field of real numbers, the set of j-planes is completely determined and j is any real number > -½