COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

A projective coqfiguration
J. W. P. HIRSCHFELD
1. In ageometry over a field, four skew lines not lying in a regulus have
two transversals (which may coincide or lie only in a quadratic extension
of the field). From this come the following theorems.
The double-six theorem (Schlafli, 1858): Given five skew lines with a
single transversal such that each set of four has exactly one further transversal,
the five lines thus obtained also have a transversal-the completing
line of the double-six.
Grace’s extension theorem (Grace, 1898): Given six skew lines with a
common transversal such that each set of five gives rise to a double-six,
the six completing lines also have a transversal-the Grace line.
Conjecture: Given seven skew lines with a common transversal such
that each set of six gives rise to a Grace figure, the seven Grace lines also
have a transversal.
2. The double-six is self-polar and lies on a unique cubic surface, which
contains 27 lines in all. The configuration exists for all fields except GF(q)
for q = 2, 3 and 5 [l].
The six initial lines in the Grace figure are chords of a unique twisted
cubic and are polar to the completing lines, which are also chords of the
cubic. However, the theorem as it stands is true only if the six completing
lines of the double-sixes are skew to one another. This is not necessarily
true, as the six lines may be concurrent. The configuration exists for GF(9)
but not for GE;(q) with q < 9 [2].
The conjecture depends only on the incidences of the lines. So, if it is
true over the complex field, it is true over any finite field large enough for
the seven Grace lines to exist.
N7ren [3] mentions that both he and Grace attempted the conjecture but
were not able to achieve anything. Grace proved his theorem by, in fact,
first establishing a slightly more general result. He proved a theorem for
linear complexes, which was then applied to special linear complexes,
which was in turn dualized to the theorem of the extension of the double-six.
In all, no one was very hopeful of extending Grace’s theorem, which
itself was regarded as something of a fluke. Further scepticism set in on
finding that the conditions of linear independence on the initial set of
lines were not even sufficient for Grace’s theorem.
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