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

[摘要] Let V denote a 5-dimensional vector space over , field F, and let (b(ij)) 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 less-than-or-equal-to i < j less-than-or-equal-to 5} for LAMBDA2V. It is well known that b (not-equal 0) is decomposable (pure, simple) if and only if its components b(ij) satisfy a set of five quadratic conditions resulting from the Grassmann relations. In the case F = GF(2) it is shown that these five quadratic conditions are equivalent to a single 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.

[发布日期] 1994-10-01