COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

322 J. W. P. Hirschfeld
For the conjecture, the field GF(31) was chosen and, using a computer,
an initial set of lines found for which the seven Grace lines existed. The
latter did not have a common transversal. Hence the conjecture is false.
3. A theoretical approach to the problem is more difficult to envisage.
In Grace’s figure, all was symmetry. There were two sets of 12 and 32
lines involved forming 32 double-sixes: each of the 32 lines met 6 of the
12 and could provide a starting point for the construction.
In the conjecture, the line b meets al, a2, a3, a4, as, as, a7 which have in
fours the further transversals b,. Then there are 21 double-sixes like
a12 a3 a4 a5 a6 a7
b bm blz4 bm blz6 bm
and 21 lines xii. The six lines aii (j + i) have a transversal Bi, the Grace
line, giving seven lines Bi. There are also seven twisted cubits ti, where ti
has the 12 chords aj, ai] (j =l= i). The seven ti have a point P in common.
The dual result to this is that there is a unique plane z meeting b and the
7 ai in a conic C. The cubits ti and tj have in common the 6 chords crij,a,
(k + i, j), which is the full complement for two twisted cubits with a point
in common.
Apart from the 71 lines so far obtained-l b, 7 ai, 35 b,, 21 xii, 7 pithere
are 105 further lines & like ,%&, the line common to the reguli
(b145bdw), (h&5&57), (h&&6i), (bdub176); there is no reason to
suppose that /$ = & = ,!Ii3. However, the five lines & (k =l= i, j) all lie
in a regulus.
Since seven lines are under discussion (both the 7 ai and the 7 &), Cayley’s
problem of seven lines lying on a quartic surface would appear
relevant. There are 34 linear conditions to determine a quartic surface
and 5 conditions for it to contain a given line. There are but 33 conditions
for b and the 7 ai to lie on a quartic surface. Hence there is a linear family
FO+ XFr of quartic surfaces through these lines. One member of the pencil,
Fl say, contains P (as a node) and hence the 7 cubits ti. Another, Fo say,
has b as a double line, so that any two members of the pencil touch along
b. The curve of degree 16 common to all the surfaces consists of b (twice),
al, a2, a3, ad, as, a& a7 plus a rational irreducible septimic S, which is
quadrisecant to each ai and unisecant to b. Both F,, and Fl are uniquely
defined.
The pencil of planes through b meets FO residually in a pencil of tonics,
eight of which break up into a pair of lines ai, a; (i = 1, . . . , 8). The conic
C lies on FO and meets one of aa, ai, which are incidentally both trisecant
to S. In this way, FO contains 27 = 128 tonics. Several special cases may
occur for this set of eight pairs of lines. Seven pairs, by the nature of the
construction, must lie in the field. The eighth pair may lie in a quadratic
extension of the field. There may be a node at the intersection of a pair of
lines (which then counts twice). Also there may occur a second isolated
A projective conjiguration 323
node collinear with the first and a point of b forming a torsal line of FO
and making a7, a;, say, coincide. This torsal line then contains three nodes,
four being required before it is a double line. The last case was in fact
the one to occur in the computed example.
In the calculated example, the 7 lines pi were not in fact quite general:
one pair had a point in common. The seven lines therefore lay on a quartic
surface, which was unique. This surface contained another three lines. It is
not at all clear if there is always a quartic surface through the 7 bi.
It is also possible for the 7 cubits ti to coincide, in which case one member
of the family of surfaces through the ai is ruled.
Attempts to connect the 7 ,& to the pencil FO+ IF1 and to obtain some
contradiction from the 176 lines mentioned above were unsuccessful.
My gratitude is to Dr. D. Barton for calculating the lines, to Mr. J. M.
Taylor for help with the further computing and to Prof. J. G. Semple and
Dr. J. A. Tyrrell for the exegesis of the pencil of quartic surfaces.
REFERENCES
1. J. W. P. HIRSCHFELD: Classical configurations over finite fields: I. The double-six
and the cubic surface with 21 lines. Rend. Mat. e Appl. 26 (1967), fast. 1-2.
2. J. W. P. H IRSCHFELD: Classical configurations over finite fields: II. Grace’s extension
of the double-six. Rend. Mat. e Appl. 26 (1967), fast. 3-4.
3. T. L. WREN: Some applications of the two-three birational space transformation.
Proc. London Math. Sm. (2) 15 (1916), 144.