THE LINES OF PG(4, 2) ARE THE POINTS ON A QUINTIC IN PG(9, 2)

Let V denote a 5-dimensional vector space over , field F, and let (b(i j)) denote the 10 independent components of a bivector b is-an-element -of LAMBDA2V relative to a choice of product basis {e(i) AND e(j): 1 l ess-than-or-equal-to i < j less-than-or-equal-to 5} for LAMBDA2V. It i s well known that b (not-equal 0) is decomposable (pure, simple) if an d only if its components b(ij) satisfy a set of five quadratic conditi ons resulting from the Grassmann relations. In the case F = GF(2) it i s shown that these five quadratic conditions are equivalent to a singl e quintic condition. In projective language the 155 lines of PG(4, 2) are therefore seen to be (in 1 - 1 correspondence with) the 155 points on a certain quintic lying in PG(9, 2). (C) 1994 Academic Press, Inc.