ABSTRACT
Indicator sets provide an alternative manner of determining spreads and were developed initially by R.H. Bruck. We begin with indicator sets which produce spreads of PG(3, q).
Consider a 4-dimensional vector space V over a field K isomorphic to GF (q). Form the tensor product of V with respect to a quadratic field extension F of K, F isomorphic to GF (q2), V ⊗K F . If we form the corresponding lattices of subspaces to construct PG(3, F ), we will have a PG(3,K) contained in PG(3, F ), such that, with respect to some basis for V over K, (x1, x2, y1, y2), for x1, x2, y1, y2 ∈ K represents a point homogeneously in both PG(3,K) and PG(3, F ). The Frobenius automorphism mapping defined by
ρq : (x1, x2, y1, y2) 7−→ (x q
1, x q
2, y q
1, y q
is a semi-linear collineation of PG(3, F ) with set of fixed points exactly PG(3,K). We use the notation Zq = (x1, x2, y1, y2)
q = (xq1, x q
2, y q
1, y q
2). Finally, choose a line PG(1,K) within the given PG(2,K) in the analogous manner so that there is a corresponding PG(1, F ). Hence, we have
PG(1,K) ⊆ PG(3,K),
PG(1,K) ⊆ PG(1, F ) ⊆ PG(2, F ).
